Design of an aggregated marketplace under …...a bid that comprises a ﬁxed price and a target...

Design of an aggregated marketplace undercongestion effects: Asymptotic analysis andequilibrium characterization

Ying-Ju Chen, Costis Maglaras, and Gustavo Vulcano

Abstract We study an aggregated marketplace where potential buyers arrive andsubmit requests-for-quotes (RFQs). There are n independent suppliers modeled asM/GI/1 queues that compete for these requests. Each supplier submits a bid thatcomprises of a fixed price and a dynamic target leadtime, and the cheapest supplierwins the order as long as the quote meets the buyer’s willingness to pay. We charac-terize the asymptotic performance of this system as the demand and the supplier ca-pacities grow large, and subsequently extract insights about the equilibrium behav-ior of the suppliers. We show that supplier competition results into a mixed-strategyequilibrium phenomenon that is significantly different from the centralized solution.In order to overcome the efficiency loss, we propose a compensation-while-idlingmechanism that coordinates the system: each supplier gets monetary compensationfrom other suppliers during his idle periods. This mechanism alters suppliers’ ob-jectives and implements the centralized solution at their own will.

Ying-Ju ChenSchool of Business and Management & School of Engineering, The Hong Kong University of Sci-ence and Technology, Clear Water Bay, Kowloon, Hong Kong, e-mail: [email protected]

Costis MaglarasColumbia Business School, 409 Uris Hall, 3022 Broadway, New York, NY 10027 e-mail:[email protected]

Gustavo VulcanoSchool of Business at Universidad Torcuato di Tella, Buenos Aires, Argentina, e-mail: [email protected]

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2 Ying-Ju Chen, Costis Maglaras, and Gustavo Vulcano

1 Introduction

1.1 Background and motivation

Electronic markets (e-markets) have proliferated in the last decade or so as means toefficiently aggregate supply and demand for services or goods, in an effort to reducesearch and transaction costs, improve market outcomes, and benefit both partici-pants that supply and demand services. We study a mathematical model motivatedby such marketplaces for services or goods that are subject to congestion effects,manifested in terms of delays until the good is delivered. The focus is on analyzingthe market dynamics and gaining insight regarding the competition across suppli-ers when services are produced in a make-to-order manner (that is modeled via aqueueing facility) and where congestion signals are state-dependent.

As part of the dynamics of the e-market evolution, several large-scale, web-based service sites have recently emerged. An incomplete list of examples includesODesk, Elance, Vworker, Freelancer.com and Guru, which are freelancing sites thatfacilitate and streamline the process of hiring virtual or remote workers. In theseplatforms, many small service providers (e.g., individual computer programmers)seek work orders. Customers (e.g., employers) post job descriptions in the form ofRFQs, and have service providers bid on the work. Customers then look at previ-ous ratings and work history of the different candidates before settling on either acontract rate, or a pay-per hour agreement. Generally, money is escrowed by eachof the websites (intermediaries), which release the payment to the service providerwhen the work is completed, while skimming a commission –typically 5-15% ofmoney that changes hands. In addition, sometimes intermediaries also charge amembership fee to the parties involved. The volume of registered freelancers andeffective transactions conducted through these platforms has been growing expo-nentially over time. Just to illustrate, the numbers of work hours per week transactedthrough oDesk were 50,000 by August 2008, 100,000 by June 2009, and 450,000 byJuly 2011.1 An important portion of the projects auctioned out via these markets arecomplex and could be better addressed via a team as opposed to an individual. Tobetter serve and bid in such cases, many freelance workers are represented via agentsthat pool capacity as traditional agents would do, and also provide project manage-ment in executing the complex projects so as to best use the pooled resources. Theseagents aggregate the capacity of many individual freelance providers.

In these settings, customers usually require specific skill sets, quality, and time-liness from their providers, and account for these needs as well as for cost in theirutility function. This multidimensional assessment can be captured by a scoring in-dex that a customer assigns to each potential provider. The bids are ranked, and theorder is then awarded to the most “desirable” service provider. In this way, the finalallocation for each work order is decided based on a reverse auction. Even thoughmultiple service attributes can be subsumed in the scoring function, two of the mostrelevant ones are expected delay and price. Customers visiting these marketplaces

1 Data available from www.odesk.com.

Design of an aggregated marketplace under congestion effects 3

usually seek quick solutions and are willing to trade prices with waiting times. In thisregard, aggregated marketplaces like the aforementioned ones raise several interest-ing practical and theoretical questions. In general, the role of the intermediary ispassive, neutral, and limits to pooling supply and demand in an exchange platform.In this case: How does the system dynamics evolve as service providers (suppliers)compete for each potential order by posting a price and a state-dependent delay esti-mate? How should these suppliers determine their bidding strategies? Is the marketefficient under competitive behavior? If the market were inefficient, then the role ofthe intermediary may become more active, in the sense of proposing a coordinationscheme to align the suppliers’ incentives. Is it possible to achieve this centralizedoptimal solution? If so, how can it be implemented in practical terms?2

We make some initial progress in addressing these questions. Specifically, weintroduce a stylized mathematical model to study the aggregated marketplace insettings characterized by high volume of transactions. The goal of our study is tounderstand the dynamics and performance of the system, and gain insight on thepricing and capacity game among suppliers. We assume that potential buyers arriveaccording to a Poisson process and submit order requests, and that the suppliers(modeled as M/GI/1 queues) compete for these requests. Initially, suppliers decidethe capacity (i.e., service rate) to offer, which is a static, long term decision. Whileoperating, each supplier processes orders in a first-in-first-out manner, and submitsa bid that comprises a fixed price and a target leadtime that depends on his ownqueue status. In fact, a key distinction of our work is that the delay quotations aredynamic rather than based on a steady-state assessment of the queue size. Whensuppliers submit bids, they face an economic tradeoff: a high price will lead to highrevenues per order, but will reduce the total number of orders awarded, which willcause excessive idleness and implicit revenue loss; a low price will result in manyawarded orders and large backlogs, that, in turn, will cause long delay quotationsthus increasing the full cost of the respective bid. The arriving buyer then uses ascoring function to compute the net utility associated with her bid, and awards theorder to the lowest-quote supplier in order to maximize her own surplus (providedthat it is nonnegative).

2 There are other applications that share the same salient feature of several firms competing inoffering some type of substitutable service that is differentiated with respect to its price and delay.Perhaps one of the most pervasive comes from the US equities market, which comprises of manyexchanges, such as the NYSE, NASDAQ, ARCA, BATS, etc. Exchanges typically function aselectronic limit order books, operating under a “price-time” priority rule, and their high-frequencydynamics can be modeled as multi-class queueing systems. Exchanges offer a rebate to liquid-ity providers, i.e., traders that post limit orders that “make” markets when their orders get filled,and charge a fee to “takers” of liquidity that initiate trades using marketable orders that transactagainst posted limit orders. The magnitudes of these make-take fees vary across exchanges and arecomparable to the spread plus a significant fraction of the overall trading costs. Exchanges oftenchange their fees and rebates in an effort to attract liquidity. Market participants employ so called“smart order routers” that take into account real-time market data, including queue and trading rateinformation, and formulate an order routing problem to trade off between rebates and a notion ofexpected delay, fill probabilities, and/or expected adverse selection. Once again, prices are fixedbut delays are state dependent.


1.2 Overview of results

This appears to be one of the first papers to study competition in queues with sub-stitutable products or services and state dependent congestion information. The dis-crete choice among substitutable products of potential consumers, the state depen-dent nature of the congestion signals, and the decentralized control among supplierscomplicate the analysis of this system, rendering brute force analysis to be essen-tially intractable.3

The first contribution is to propose a tractable way for studying the decentralizedmarket. As a preliminary step towards solving the capacity and pricing game, weanalyze the queueing performance of this stochastic dynamic system assuming thatthe price vector is given. The solution to this problem allows suppliers to evaluatetheir revenues given their prices as well as the prices of the competitors, which isan essential subroutine in the equilibrium analysis of the supplier pricing game. So,given a specified price vector, our first set of results characterizes the behavior ofthe marketplace using an asymptotic analysis where the potential demand and thesupplier processing capacities grow large simultaneously. This asymptotic analysisis motivated by the following observation: If this market were served by a uniquesupplier (modeled as an M/GI/1 queue as well), then it would be economicallyoptimal for this supplier to set the price that induces the so-called “heavy-traffic”operating regime; i.e., rather than assuming that the system is operating in the heavytraffic regime, as is often done, this result provides a primitive economic foundationthat this regime emerges naturally since it optimizes the system-wide revenues (e.g.,see [7] and [19]). Specifically, if Λ is the market size, then the above result statesthat the economically optimal price is of the form p∗ = p+π/

√Λ , where p is the

price that induces full resource utilization in the absence of any congestion, and π isa constant.

With the above fact in mind, we formulate the performance analysis sub-problemin a novel way that becomes asymptotically tractable in settings with large capaci-ties and large volume of transactions. Specifically, based on the above observation,the starting point of our analysis is to write the suppliers’ price bids as perturba-tions around the price p of the form pi = p+ πi/

√Λ , for a constant πi, where Λ

is viewed as a natural proxy for system size. Letting Λ grow large, we derive thecorresponding fluid and diffusion approximations. The fluid model transient analy-sis is helpful in establishing an important state space collapse (SSC) result througha variation of an approach developed by [8]. The SSC property establishes that thesuppliers’ dynamics are asymptotically coupled and can be described as a function

3 The distinction regarding consumer choice model is important. With partially substitutable prod-ucts, one could model consumer choice through a multi-product demand function, where the de-mand for one product depends on the price and delay of all products in a continuous manner. Thisis not the case with perfectly substitutable products, where demand may switch from one productto another in a discontinuous manner. For example, in a setting with two equally priced products,all consumers will select the one with the lower delay. This would increase the delay estimate ofthe faster option, causing all consumers to choose the alternative option. That is, small differencesin price and delay, may lead to dramatic differences in demand for one of suppliers.


of the aggregate (system-wide) workload process, and, moreover, SSC implies thata supplier is able to know his competitors’ bids by simply observing awaiting or-ders in his own buffer. We prove a weak convergence result of the workload processto a one-dimensional reflected Ornstein-Uhlenbeck (O-U) process, where interest-ingly the reflection point may be away from zero depending on the suppliers’ prices.The latter implies the nonintuitive property that the aggregate workload process cannever drain even though some of the suppliers may be idling, and this happens if thesuppliers differ in their pricing. The derivation of the diffusion model extends “stan-dard” results to this setting with self-interested routing policies based on dynamicinformation, which is of independent interest.

Second, using the asymptotic characterization of system behavior at a fixedchoice of prices and capacities, we characterize the revenue stream of each sup-plier using the steady state properties of the reflected O-U process. This is then usedto study the resulting pricing game. We find that the pricing game does not admit apure strategy equilibrium. We specify the structure of the supporting mixed strategyequilibria where suppliers randomize over their pricing decisions. We also provethat the second-order efficiency loss of the decentralized solution can be arbitrarilylarge. It is worth noting that our approximate analysis of the supplier game is inter-nally consistent in the sense that the lower order price perturbations that essentiallycapture the supplier pricing game do not become unbounded, but rather stay finite.In essence, all suppliers choose to operate in the asymptotic regime we identifiedand used in our analysis. The framework of studying the appropriate asymptoticformulation of the aggregated market in the context with self interested buyers andstate dependent congestion information, and using the derived diffusion to study thesupplier game is novel. Such problems had not been studied in the literature be-fore, in part due to their inherent complexity, and their proposed roadmap advancedseems to be of broader interest.

Third, the discrepancy between the centralized solution and the decentralizedequilibrium calls for the development of a mechanism to coordinate the marketplace,in the sense that all suppliers would self-select to price according to the centralizedsolution. Our proposal relies on a transfer pricing scheme that compensates suppliersduring idle periods. The existence of the intermediary in the motivating examplesdescribed above provides the natural support to implement it.

1.3 Literature review

Our work touches on three related bodies of literature: 1) Economics of queues,2) Competitive models in queueing contexts, and 3) Approximation schemes to an-alyze complex queueing models.

The literature that studies pricing in the context of single-server queues datesback to [23]. The demand model that we consider here is inspired by [21]: Thereis a single class of potential customers that arrive according to a Poisson arrivalprocess, each having a private valuation that is an independent draw from a general


distribution, and a delay sensitivity parameter that is common across all customers.[22] extends that model to multiple customer types, and [1] extends it to a revenuemaximization setting. In the context of queueing models with pricing and servicecompetition, starting from the early papers by [16] and [18], customers are com-monly assumed to select their service provider on the basis of a “full cost” thatconsists of a fixed price plus a waiting cost. In both [16] and [18], competition ismodeled in a duopoly setting where firms operate as M/M/1 queueing systems. Re-laxations of the early papers include [17], which studies a variant of [18] in whichthe providers are modeled as symmetric M/GI/1 systems. [15] generalizes [17] forarbitrary number of service providers. [3] treats the price and waiting time cost asseparate firm attributes that can be traded off differently by each arriving customer.These papers focused on customers making decisions based on steady-state perfor-mance measures.

Our use of asymptotic approximations and heavy traffic analysis to study thesupplier game is motivated by the results of [7] and [19], who showed that in largescale systems the heavy traffic regime is the one induced by the revenue maximiz-ing price. Our work implicitly assumes the validity of the heavy traffic regime inderiving its asymptotic approximation (as opposed to proving it as in the two papersabove). The equilibrium pricing behavior of the competing suppliers supports therationale of this assumption in the sense that no supplier wishes to price in a waythat would deviate from that operating regime. The derivation of our limit modelmakes heavy use of the work by [20] on queues with state dependent parameters,and of the framework developed by [8] for proving state space collapse results. Wealso use technical results from [5] and [26] in our analysis. However, the combina-tion of all our model features does not fit neatly the technical requirements of theaforementioned papers, as shown in the proofs contained in the online appendix.

A queueing paper that studies a model that is similar to our in a heavy trafficasymptotic regime is [24]. The key differences are the following: a) [24] assumesstrictly convex delay cost functions as opposed to linear, b) it does not considerpricing (or some term that could account for its effect in the routing rule), and c)it does not allow for admission control decisions that can turn away users whenthe system is congested. The latter is captured by the behavior of self-interestedusers that differ in their valuations, and as a result will choose not join the market ifthe full cost exceeds their value. Taken together, these three elements necessitate anew analysis that leads to some insights that differ than what was observed in [24].Perhaps the most notable difference is the fact that as a result of the pricing game,the workload process will not reflect at the origin, but instead it will reflect at somestrictly positive quantity.

Recent papers accounting for congestion pricing include [11], which studies twotypes of contractual agreements in oligopolistic service industries. [4] appeals to anasymptotic analysis to study a competitive game of a queueing model, and proposea general recipe for relating the asymptotic outcome to that of the original system.They show that the pricing decisions and service level guarantees result in respec-tively first-order and second-order effects on the suppliers’ payoffs. More recently,[2] studies a large-scale marketplace with a moderating firm and numerous service


providers. [2] also uses a static measure of the waiting time standard (usually theexpected value or some percentile of the steady state distribution).

The remainder is organized as follows. In §2, we describe the model details.Next, §3 derives the asymptotic characterization of the marketplace behavior, and§4 characterizes the equilibrium behavior of the supplier pricing game. Finally, §5includes our concluding remarks. All the proofs and more details of the analysis canbe found in the technical report [10].

2 Model

2.1 Description of the market

We consider an aggregated marketplace where a homogeneous product (e.g., com-puter programming hours) is exchanged. The market functions as follows:Order arrivals: Potential buyers arrive to the marketplace according to a Poissonarrival process with intensity Λ , and submit requests-for-quotes (RFQs). Each RFQcorresponds to the procurement of one unit of the product. Each buyer has a privatevaluation v for her order that is an independent draw from a general, and continu-ously differentiable distribution F(·). Buyers are delay sensitive and incur a cost cper unit of delay. Thus, buyers are homogeneous with respect to delay preferences,and heterogeneous with respect to valuations (though symmetric across the commonc.d.f. F(·)). A buyer that arrives at time t initiates a RFQ process to procure one unitof the product.Suppliers: The market is served by a set of suppliers N = 1, . . . ,n. Each supplieri is modeled as an M/GI/1 queue with an infinite capacity buffer managed in a First-In-First-Out fashion. Service times at supplier i follow a general distribution withmean 1/µ

′i and standard deviation σi. Let µ := ∑i∈N µ

′i/Λ be the (normalized) ag-

gregated service rate of the market. We assume that the capacity vector µ′ ≡ µ ′i’s

is common knowledge. This fact can also be sustained by information provided byintermediary entities like the ones discussed in §1.Market mechanism: Suppliers compete for this request by submitting bids that com-prise a price pi and a target leadtime di(t). We assume that the price component ofthe bid is state-independent, i.e., supplier i always submits the same price bid pi forall orders. The leadtime component of the bid submitted by each supplier i is state-dependent and equals the expected time it would take to complete that order; cf. (6)later on. We are assuming here that the supplier always submits a truthful estimateof the expected delay di(t). In fact, in the presence of a market intermediary, themisreport of expected delays is discouraged through the display of past experiencesof buyers with a given supplier, e.g. through publicly available ratings and reviews.This revealed information acts as a threatening device to favor the honest disclosureof suppliers’ availabilities.


On their end, buyers are price and delay sensitive, and for each supplier i theyassociate a “full cost” given by pi+cdi(t), where the delay sensitivity parameter c isassumed to be common for all buyers. Upon reception of the bids, the buyer awardsher order to the lowest cost supplier, provided that her net utility is positive, i.e.,v≥mini∈N pi+cdi(t); otherwise, the buyer leaves without submitting any order.Whenever a tie occurs, the order is awarded by randomizing uniformly among thecheapest suppliers.4

Given vectors p = (p1, . . . , pn), and d(t) = (d1(t), . . . ,dn(t)), the instantaneousrate at which orders enter this aggregated market is given by

λ (p,d(t)) = Λ F(minipi + cdi(t)). (1)

Focusing on the right-hand-side of the above expression, we note that the buyers’valuation distribution F(·) determines the nature of the aggregate demand rate func-tion.5 Let x = mini pi and, with slight abuse of notation, write λ (x) in place of

λ (p,0), where 0 is the vector of zeros. We further define ε(x) =−dλ (x)dx

xλ (x)

. The

expression ε(x) can be regarded as the price elasticity of the demand rate, as it mea-sures the proportional change of demand rate in response to the price change. Wewill make the following intuitive economic assumption:

Assumption 1 λ (p,0) is elastic in the sense that ε(x)> 1 for all price vectors p inthe set p : 0≤ λ (p,0)≤ ∑

ni=1 µ

′i and x = mini pi.

The above assumption implies that in the absence of delays, a decrease in theminimum price would result in an increase in the market-wide aggregated revenuerate p ·λ (p,0).6 Of course, this would increase the utilization levels of the suppli-ers and lead to increased congestion and delays, thereby moderating the aggregatearrival rate λ (p,d(t)).

Let A(t) be the cumulative number of orders awarded to all the competing sup-pliers up to time t,

A(t) = N(

Λ

∫ t

0F(min

i∈Npi + cdi(s))ds

), (2)

4 We could also allow other tie-breaking rules, and it can be verified that our results are not proneto the specific choice of tie-breaking rules.5 For example, if v ∼U [0,Λ/α], then the demand function is linear, of the form λ (x) = Λ −α x,where x = mini pi + cdi(t); if v∼ Exp(α), then the demand is exponential, with λ (x) = Λ e−αx.6 This implication follows directly from the economics literature. When ε(x)> 1, the proportionalincrease of demand rate is larger than the proportional decrease of price. As the revenue is theproduct of price and demand rate, the aggregated revenue rate ends up being higher. Supposefurther that there are no congestion effects, there exists a central planner that could select a commonprice p and an aggregate capacity µ. Under a linear capacity cost hµ and the arrival rate Λ , thesolution to the problem maxp,µ pλ (p,0)− hΛ µ : 0 ≤ λ (p,0) ≤ Λ µ results in a capacitydecision that satisfies the above assumption.


where N(t) is a unit rate Poisson process and the equality holds only in distribution.To represent the cumulative number of orders for each individual supplier, let

J (t)≡ i ∈N : pi + cdi(t)≤ p j + cd j(t),∀ j ∈N (3)

as the set of cheapest suppliers at time t. Further, define ΞJ (t) as the randomvariable that assigns the orders uniformly amongst the cheapest suppliers. That is,

ΞJ (t) = i with probability1

|J (t)|if i ∈J (t), where |J (t)|> 0 is the cardinality

of J (t), and ΞJ (t) = i with zero probability otherwise. For the ease of the exposi-tion, we will assume that J (t) and ΞJ (t) are defined as continuous processes forall t ≥ 0, even if no actual arrival occurs at that time. This allows us to write thecumulative number of orders awarded to supplier i, denoted by Ai(t), as

Ai(t) =∫ t

011ΞJ (s) = idA(s), (4)

where 11· is the indicator function. Note also that A(t) = ∑i∈N Ai(t).Supplier dynamics: Let Qi(t) denote supplier i’s number of jobs in the system (i.e.,in queue or in service) at time t, and Ti(t) denote the cumulative time that supplieri has devoted into producing orders up to time t, with Ti(0) = 0. Let Yi(t) denotethe idleness incurred by supplier i up to time t. Note that Ti(t)+Yi(t) = t for eachsupplier i; moreover, Yi(t) can only increase at a time t when the queue Qi(t) isempty. Let Si(t) be the number of supplier i’s service completions when workingcontinuously during t time units, and Di(t) = Si(Ti(t)) be the cumulative number ofdepartures up to time t. The production dynamics at supplier i are summarized inthe expression:

Qi(t) = Qi(0)+Ai(t)−Di(t). (5)

Given this notation, then

di(t) =Qi(t)+1

µ′i

, (6)

is the expected sojourn time of the new incoming order, given that it gets awarded tosupplier i and the current queue length is Qi(t). Under our modeling assumptions,supplier i will therefore bid (pi,di(t)), where di(t) is given by (6). Each supplierknows his own system queue length Qi(t), but is not informed about his competitors’queue lengths.

2.2 Problems to address

We study three problems related to the market model described above:

1. Performance analysis for a given p and µ′: Given a fixed price vector p and

a vector of processing capacities µ′, the first task is to characterize the system

performance, i.e., to characterize the behavior of the queue length processes Qi(t)


at each supplier, and calculate the resulting revenue streams for each supplier. Asupplier’s long-run average revenue is

Ωi(pi, p−i)≡ pi · limt→∞

Si(T (t))t

, (7)

where p−i ≡ (p1, . . . , pi−1, pi+1, . . . , pn) denotes other suppliers’ price decisions.Our goal, therefore, is to analyze the performance of relevant system dynamicsthat leads to a tractable representation of these long-run average revenues.

2. Characterization of market equilibrium: The above problem serves as an inputto study the competitive equilibrium that characterizes the supplier capacity andpricing games, both of which are one-shot games where each supplier selectshis service rate and static price, successively. We further show that the capac-ity selections constitute the first-order effects on the suppliers’ payoffs and thepricing decisions are of second order; thus, we can conveniently decouple theequilibrium analysis into two separate stages. For the first-stage, capacity game,since the capacities (service rates) are assumed to be publicly observable, weadopt the Nash equilibrium as our solution concept. For the second-stage, pricinggame, the suppliers may be uninformed about the queue lengths of the competi-tors, which may potentially lead to information incompleteness. However, as wewill show, the suppliers’ competitive behavior is insensitive to this knowledge;consequently, we adopt again the standard Nash equilibrium (under complete in-formation) as our solution concept. Given the revenue specified in (7), a Nashequilibrium p∗i requires that p∗i = argmaxpi Ωi(pi, p∗−i), ∀i ∈N .

3. Market efficiency and market coordination: Our objective here is to characterizethe efficiency loss:

maxp

∑

i∈NΩi(pi, p−i)

− ∑

i∈NΩi(p∗i , p∗−i), (8)

i.e., the difference between the revenue of a system where a central planner wouldcontrol the pricing decision of each supplier (i.e., the first best solution), and thesum of the revenues collected in the competitive framework. If the market equi-librium is inefficient, we would like to specify a simple market mechanism thatcoordinates the market and achieves the first best solution identified above. Sucha mechanism could specify, for example, the rules according to which orders areallocated and payments are distributed among the suppliers.

3 Asymptotic analysis of marketplace dynamics

This section focuses on the first problem described in §2. Despite the relatively sim-ple structure of the suppliers’ systems and the customer/supplier interaction, it isstill fairly hard to study their dynamics due to the state-dependent delay quotations


and the dynamic allocation of orders. Our approach is to develop an approximatemodel for the market dynamics that is rigorously validated in settings where thedemand volume and processing capacities of the various suppliers are large. In thisregime, the market and supplier dynamics simplify significantly, and are essentiallycaptured through a tractable one-dimensional diffusion process. This limiting modelprovides insights about the structural properties of this market, and provides a vehi-cle within which we are able to analyze the supplier game and the emerging marketequilibrium. This is pursued in the next section.

3.1 Background: Revenue maximization for an M/M/1monopolistic supplier

As a motivation for our subsequent analysis, this subsection will summarize someknown results regarding the behavior of a monopolistic supplier modeled as anM/M/1 queue that offers a product to a market of price and delay sensitive cus-tomers. The supplier posts a static price and dynamically announces the prevailing(state-dependent) expected sojourn time for orders arriving at time t, which is givenby d(t) = (Q(t)+ 1)/µ . The assumptions on the customer purchase behavior arethose described in the previous section. Given p and d(t), the instantaneous demandrate into the system at time t is given by λ (t) = Λ F(p+ cd(t)). The supplier wantsto select p to maximize his long-run expected revenue rate.

It is easy to characterize the structure of the revenue maximizing solution insettings where the potential market size Λ and the processing capacity µ grow large.Specifically, we will consider a sequence of problem instances indexed by r, whereΛ r = r and µr = rµ; that is, r denotes the size of the market. The characteristics ofthe potential customers, namely their price sensitivity c and valuation distributionF(·), remain unchanged along this sequence. Let p = argmax pF(p), and p be theprice such that relation Λ rF(p) = µr holds. That is, neglecting congestion effects,p is the price that maximizes the revenue rate and p is the price that induces fullresource utilization, and both of these quantities are independent of r. Assumption 1implies that p < p (or equivalently, Λ rF(p) > µr) thus accentuating the tensionbetween revenue maximization and the resulting congestion effects. [7] showed thatthe revenue maximizing price, denoted by p∗,r, is of the form

p∗,r = p+π∗/√

r+o(1/√

r), (9)

where π∗ is a constant independent of r. Moreover, the resulting queue lengths Qr(t)are of order

√r, or in a bit more detail, the normalized queue length process Qr(t) =

Qr(t)/√

r has a well defined stochastic process limit as r→∞. Since the processingtime is itself of order 1/r, the resulting delays are of order 1/

√r. Following [19],

the delays are moderate in absolute terms (of order 1/√

r) but significant whencompared to the actual service time (of order 1/r). If the supplier can select theprice and capacity µ , the latter assuming a linear capacity cost, then the optimal


capacity choice is indeed such that p < p (where p is determined by µ), i.e., makingthe above regime the “interesting” one to consider. Finally, we note that the aboveresults also hold for the case of generally distributed service times ([7]).

3.2 Setup for asymptotic analysis

Given a set of suppliers characterized by their prices and capacities pi,µ′i , we pro-

pose the following approximation:

1. Define the normalized parameters µi = µ ′i/Λ for every supplier i.2. Define p to be the price such that Λ F(p) = ∑i∈N µ ′i . Define πi =

√Λ(pi− p) so

that the prices pi can be represented as pi = p+πi/√

Λ .3. Embed the system under consideration in the sequence of systems indexed by r

and defined through the sequence of parameters:

Λr = r, µ

ri = rµi, ∀i ∈N , cr = c, v∼ F(·), (10)

and prices given by pri = p+πi/

√r for all i.

Given the preceding discussion, one would expect that the market may operate ina manner that induces almost full resource utilization, and where the underlying setof prices takes the form assumed in item 3) above. This would be true if the marketwere managed by a central planner that could coordinate the supplier pricing andcapacity decisions. The approach we pursue is to embed the system we wish to studyin the sequence of systems indexed by r and described in (10), and subsequentlyapproximate the performance of the original system with that of a limit system thatis obtained as r→ ∞, which is more tractable. Note that for r = Λ in (10), whereΛ denotes the market size of the potential order flow as described in the previoussection, we recover the exact system we wish to study. If Λ is sufficiently large, thenthe proposed approximation is expected to be fairly accurate.

The remainder of this section derives an asymptotic characterization of the per-formance of a market that operates under a set of parameters (p,µ ′) that are embed-ded in the sequence (10).

3.3 Transient dynamics via a fluid model analysis

The derivation of the asymptotic limit model (specifically, Proposition 2) will showthat the following set of equations

Qi(t) = Qi(0)+ Ai(t)− Di(t),∀i ∈N , (11)

Ai(t) =∫ t

0

1|J (t)|

11i ∈J (t)Λ F(p)ds,∀i ∈N , (12)


Di(t) = µiTi(t),∀i ∈N , (13)∫ t

0Qi(s)dYi(s) = 0,∀i ∈N , (14)

Ti(t)+ Yi(t) = t,∀i ∈N , (15)

W (t) = ∑i∈N

Qi(t)µi

. (16)

captures the market’s transient dynamics over short periods of length 1/√

r. Thissubsection studies the transient evolution of (11)-(16) starting from arbitrary initialconditions.

In (11)-(16), the processes Q, A, D, T ,Y are the fluid analogues of Q,A,D,T,Ydefined in §2, and W is the fluid analog of the system workload W . Equation (11)keeps track of the queue sizes. Equation (12) indicates how the arrivals are routedto these servers: An arrival walks away if her valuation is sufficiently low; oth-erwise, she joins server i based on the routing rule specified in §2. From Equa-tion (12), the orders get awarded to the various suppliers at a rate Λ F(p) =∑i∈N µ ′i ,i.e., F(p) = ∑i∈N µi (as indicated by the aggregate counting process N(F(p)t)).Equation (14) demonstrates the non-idling property: Yi(t) cannot increase unlessQi(t) = 0. Equation (15) is a time-balance constraint. Finally, Equation (16) estab-lishes the connection between the total workload and the queue lengths.

The next proposition establishes that starting from any arbitrary initial condition,the transient evolution of the market (as captured through (11)-(16)) converges toa state configuration where all suppliers are equally costly in terms of the full costof the bids given by (price + c × delay). This is, of course, a consequence of themarket mechanism that awards orders to the cheapest supplier(s), until their queuelengths build up so that their full costs become equal. Simultaneously, expensivesuppliers do not get any new orders and therefore drain their backlogs until theircosts become equal to that of the cheapest suppliers. From then onwards, orders aredistributed in a way that balances the load across suppliers. This result is robust withrespect to the tie-breaking rule that one may apply when multiple suppliers share thesame full cost.

Proposition 1 Let Q, A, D, T , Y ,W be the solution to (11)-(16) with max|Q(0)|, |W (0)|≤M0 for some constant M0. Then for all δ > 0, there exists a continuous functions(δ ,M0)< ∞ such that

maxi, j∈N

∣∣∣∣∣(

πi + cQi(s)

µi

)−(

π j + cQ j(s)

µ j

)∣∣∣∣∣< δ , ∀s > s(δ ,M0). (17)


3.4 State-space collapse and the aggregate marketplace behavior

The next result shows that the transients studied above appear instantaneously in thenatural time scale of the system, and as such the marketplace dynamics evolve as ifall suppliers are equally costly at all times.

We use the superscript r to denote the performance parameters in the r-th sys-tem, e.g., Ar

i (t), Sri (t), T r

i (t), and Qri (t). The (expected) workload (i.e., the time

needed to drain all current pending orders across all suppliers) is defined as W r(t) =

∑i∈NQr

i (t)µr

i.

Motivated by the discussion in §3.1, we will optimistically assume (and later onvalidate) that the supplier queue lengths are of order

√r, and accordingly define the

re-scaled queue length processes for all suppliers according to

Qri (t) =

Qri (t)√

r. (18)

The corresponding re-scaled expected workload process is given by W r(t)=√

rW r(t)=

∑i∈NQr

i (t)µi

.

Define Zr(t) = π + cW r(t), where π = ∑i∈N πi/n and c = c/n. Zr(t) can beregarded as a proxy for the average of the second-order terms of suppliers’ bids

since Zr(t) =1n ∑

i∈N

(πi + c

Qri (t)µi

). Note that the first-order term, p, is common

for all suppliers, and can be omitted while comparing suppliers’ bids.

Proposition 2 (STATE SPACE COLLAPSE) Suppose πi + cQr

i (0)µi

= π + cW r(0) in

probability, ∀i ∈N . Then, for all τ > 0, for all ε > 0, as r −→ ∞,

P

sup

0≤t≤τ

maxi, j∈N

∣∣∣∣∣(

πi + cQr

i (t)µi

)−

(π j + c

Qrj(t)

µ j

)∣∣∣∣∣> ε

−→ 0,

P

sup

0≤t≤τ

maxi∈N

∣∣∣∣∣(

πi + cQr

i (t)µi

)− Zr(t)

∣∣∣∣∣> ε

−→ 0.

The proof applies the “hydrodynamic scaling” framework of [8], which is intro-duced in the context of studying the heavy-traffic asymptotic behavior of multi-classqueueing networks. Our model falls outside the class of problems studied in [8], butas we show in the online appendix, his analysis can be extended to address oursetting in a fairly straightforward manner.


3.5 Limit model and discussion

Proposition 2 shows that the supplier behavior can be inferred by analyzing an ap-propriately defined one-dimensional process Zr(t) that is related to the aggregatedmarket workload. This also implies that although each supplier only observes hisown backlog, he is capable of inferring the backlog (or at least the full cost) of allother competing suppliers.

The next theorem characterizes the limiting behavior of the one-dimensional pro-cess Zr(t), and as a result also those of W r(t) and Qr(t).

Theorem 1. (WEAK CONVERGENCE) Suppose πi+cQr

i (0)µi

= π+ cW r(0) in prob-

ability, ∀i ∈ N . Then Zr(t) weakly converges to a reflected Ornstein-Unlenbeckprocess Z(t) that satisfies

Z(t) = Z(0)− γc∫ t

0Z(s)ds+U(t)+

c√

σ2 + µ

µB(t), (19)

where B(t) is a standard Brownian motion, U(0) = 0, U(t) is continuous andnondecreasing, and U(t) increases only when Z(t) = π . The parameters are γ =

f (p)/F(p), and σ ≡√

∑i∈N σ2i . In addition, W r(t)⇒ 1

c(Z(t)− π), and Qr

i (t)⇒µi

c(Z(t)−πi), ∀i ∈N .

The process U(t) is the limiting process of U r(t)≡ cµ

∑i∈N

µiY ri (t) (defined in the

proof of Theorem 1), which can be regarded as the aggregate market idleness of thesystem.

This theorem characterizes the limiting marketplace behavior under a given pricevector p. The market exhibits a form of “resource pooling” across suppliers. Giventhat Z(t)≥ π , it follows that W (t)≥ 1

c maxi∈N (πi− π) := ζ . This says that unlessall the suppliers submit the same price bid, the aggregate workload in the market-place will always be strictly positive and at a given time t, some suppliers will neverincur any idleness. The intuition for this result is the following. When the queueof the most expensive supplier(s) gets depleted, and this supplier(s) starts to idle,the imbalance between the aggregate arrival rate and service rate force suppliers tobuild up their queue lengths instantaneously. Consequently, suppliers that price be-low π never deplete their queue lengths asymptotically. γ, that controls the speedof the reversion of the aggregate workload process, is extracted from the customervaluation distribution. γ measures the sensitivity of the demand function to changesto the full price ”π + cd(t)”, and it is proportional to the demand elasticity at p.

To summarize, in the limit model, the suppliers’ queue length processes followfrom Qi(t) =

µi

c(Z(t)−πi), ∀i ∈N , where Z(t) is defined through (19). The next

section will use this result as an input to study the suppliers’ pricing game.


3.6 A numerical example

To demonstrate the system dynamics, we consider a system with two M/M/1servers, delay sensitivity parameter c = 0.5, and arrival rate of buyers Λ = 1.The valuation v of each customer is assumed to follow an exponential distribu-tion with mean 0.1. The aggregate and individual service rates are respectivelyµ = e−1.3, µ1 = 0.8µ, and µ2 = 0.2µ . Moreover, suppose the price parameters are

π1 =−1,π2 =−2, and therefore π =π1 +π2

2=−1.5.

We run simulations using Arena (a discrete-event simulation software) and atplaces supplement it with Matlab to compute the relevant parameters. The Arenamodel is illustrated in the online appendix.

In Figures 1 and 2, we illustrate the workload trajectories for r = 30 and r = 80,respectively, for one replication. Given these parameters, the respective boundariesare 0.365 and 0.224 for r = 30 and r = 80. Note that our mathematical statement isestablished on the steady-state workload process. We can either perform a very longrun (say with 600,000 arrivals in expectation) and break each output record from the(single) run into a few large batches. Alternatively, we can run many replications andidentify appropriate warm-up and run-length times (for example 3,000 replications,each of which generates roughly 200 arrivals). We find that both approaches leadto very similar statistical outcomes, and hence we report only the former. Whenr = 80 and the simulation time is 600,000/80 units, we find that 92.93% of time thesystem workload is above the boundary 0.224. When r = 30 (and the correspondingsimulation time is 600,000/30 units), however, this proportion goes down to 71.22%.This suggests that our mathematical result of asymptotically negligible time belowboundary is more applicable when the scaling factor is large.

0 20 40 60 80 100 120 1400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time

Wor

kloa

d W

r (t)

LowerBound

r=30

Fig. 1 An instance of workload process when r =30.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Wor

kloa

d W

r (t)

r=80

Fig. 2 An instance of workload process when r =80.


4 Competitive behavior and market efficiency

In this section, we discuss the equilibrium behavior of suppliers in the capacity andpricing games described in §2. We use the performance characterization of §3 andcast the suppliers’ prices as “small” deviations around the market clearing price p.This distinction results in a first-stage capacity game that affects the first-order rev-enues, and a second-stage, pricing game, that adjusts prices around the first-orderprice. Then, we briefly discuss the centralized solution that maximizes the aggregatepayoffs. Next, we characterize the non-cooperative behavior of suppliers under thecompetitive environment. Finally, we propose a coordination scheme that achievesthe aggregate payoff under the centralized solution, and describe how this coordina-tion scheme can be implemented in the original system.

4.1 Suppliers’ first-order payoffs and the capacity game

Let Rri (t) denote supplier i’s cumulative revenue. Since the pricing is static, supplier i

earns Rri (t) = (p+

πi√r)Sr

i (t−Y ri (t)), where Sr

i (·) is the counting service completion

process, and Y ri (t) is the cumulative idleness process for supplier i. The next lemma

shows that the “first-order” revenues of the suppliers only depend on the first-orderprice term p and the service rates µi’s.

Lemma 1. Rri (t)r → pµit, as r→ ∞,∀i ∈N .

Lemma 1 demonstrate that the capacity choice (µi) has a first-order effect on thesuppliers’ revenues. If we attach an appropriate capacity cost ci(µi) to the suppli-ers, the capacity game can be explicitly posted. In the centralized system, a centralplanner decides the service rates (capacities) to maximize the net revenue:

maxµi

∑

iµi p(∑

iµi)−∑

ici(µi)

. (20)

This centralized solution can be obtained via a two-stage problem in which we firstfind the optimal allocation for each individual to minimize the aggregate cost:

C(µ) = minµi

∑

ici(µi),s.t. ∑

iµi = µ, µi ≥ 0, ∀i

, (21)

and then optimize over the aggregate service rate via maxµµ p(µ)−C(µ)| µ ≥ 0.In a Nash equilibrium µ∗i ’s, supplier i chooses a capacity such that

µ∗i = argmax

µi

µi p(µi +∑

j 6=iµ∗j )− ci(µi)

, (22)


where p = (F)−1(∑i µi/λ ) is the price that induces the full resource utilization.Furthermore, if µ p(µ) is concave in µ and ci(µi) is convex in µi, ∀i,7 there exists apure-strategy Nash equilibrium in the capacity game. Based on this existence result,we can characterize the pure-stra tegy equilibrium from the best responses of sup-pliers against others’ strategies. Specifically, a Nash equilibriumµ∗i n

i=1 satisfies

µ∗i p′(∑

iµ∗i )+ p(∑

iµ∗i )− c

′i(µ∗i ) = 0, ∀i. (23)

It can be verified that in the decentralized (Nash) equilibrium, each supplierintends to build a capacity µ∗i higher than the centralized solution. This over-investment result follows from the ignorance of the negative externality a supplierbrings to the entire system, as a supplier may benefit from over-investment sincethis allows him to capture a higher market share. This is reminiscent of the demand-stealing effect in the classical Cournot competition.

4.2 Suppliers’ second-order payoffs and the pricing game

To study the suppliers’ pricing game we will focus on the second order correction

around Rri (t) defined as rr

i (t)≡1√r(Rr

i (t)− r pµit),∀i ∈N . The limiting processes

of these corrected terms are characterized in the following lemma.

Lemma 2. rri (t)⇒ ri(t), as r → ∞, ∀i ∈ N , where ri(t) := µiπit + pσiBs,i(t)−

µi pYi(t) and Yi(t) is the limiting process of Y ri (t) as r→ ∞.

Instead of using the revenue functions Ωi(pi, p−i)’s defined in (7), we willstudy the suppliers’ pricing game based on their (second-order) revenues given by

Ψi(πi,π−i)≡ limt→∞

ri(t)t

= µi(πi− pE[Yi(∞)]), (24)

where π−i≡ (π1, . . . ,πi−1,πi+1, . . . ,πn), and (with some abuse of notation) E[Yi(∞)] :=limt→∞

Yi(t)t . Define hi as the (steady-state) proportion of the market idleness in-

curred by supplier i, i.e.,E[Yi(∞)] = hiE[U(∞)], (25)

where we again denote by E[U(∞)] := limt→∞U(t)

t the long-run average of theaggregate idleness U(t) specified in Theorem 1.

Dividing (19) by t and letting t→ ∞, we obtain that

E[U(∞)] = limt→∞

U(t)t

= γcE[Z(∞)] = γcβφ(π/β )

1−Φ(π/β ), (26)

7 These assumptions are commonly adopted in revenue management, although in some context thearrival rate is used instead of the service rate, which makes no difference in heavy traffic regime,see e.g. [13].


where

β =

√c(µ +σ2)

2γ µ2 , (27)

and the closed-form expression follows from the fact that the reflected Orstein-Uhlenbeck process Z(t) has the stationary distribution as a truncated Normal ran-dom variable ([9, Proposition 1]).8 We do not derive the closed-form expressions ofhi’s since they are not needed for our equilibrium analysis.

Let J = j|π j = π, where π ≡maxi∈N πi, denote the set of the most expensivesuppliers (allowing for ties). From Theorem 1 we have that Z(t) ≥ π for all t ≥ 0

and that Qrj(t)⇒

µ j

c(Z(t)−π j)> 0, for all t ≥ 0, ∀ j /∈ J. It follows that Yi(t) = 0 for

all t ≥ 0, and therefore h j = 0, ∀ j /∈ J. Given (24) and (25), we obtain the suppliers’second-order long-run average revenue functions as follows:

Ψi(πi,π−i) =

µiπi−µi phiγcβ

φ(π/β )1−Φ(π/β ) , i f i ∈ J,

µiπi, otherwise.(28)

4.3 Centralized system performance

In the centralized version of the system, a central planner makes the price decisionsπ ≡ (π1, . . . ,πn) in order to maximize the total aggregated revenue:

maxπ

∑

i∈Nµiπi− pγ µβ

φ(π/β )

1−Φ(π/β ): πi ≤ π

, (29)

where we have applied ∑i∈J µihi =µ

c to combine all the penalties imposed on themost expensive suppliers.

For convenience, we define L (π)≡ pγ µβφ(π/β )

1−Φ(π/β )as the revenue loss that

the system suffers if π is the highest price offered. The optimal pricing decisions aresummarized in the following lemma:

Lemma 3. In a centralized system, all prices πi’s are equal. The optimal static priceis πC := argmaxπ [µπ−L (π)].

8 We can verify that using their notation, the Z(t) process corresponds to the following parameters:a = γc,m = 0, and the process has only a left reflecting barrier π .


4.4 Competitive equilibrium

In a decentralized (competitive) system, each supplier is maximizing his own payoff,Ψi(πi,π−i), in a non-cooperative way: maxπi Ψi(πi,π−i). Recalling the definition ofL (π), we can rewrite the supplier’s payoff in (28) as

Ψi(πi,π−i) = µiπi−µihicµ

L (π)11i ∈ J] (30)

In the following we characterize the equilibrium behavior. We will split our discus-sion in two cases, depending on whether suppliers are endowed with homogeneousor heterogeneous service rates.

4.4.1 Homogeneous service rate case

We first consider the case where the service rates are the same across suppliers,i.e., µi = µ j ≡ µ, ∀i, j ∈ N , and focus on symmetric equilibria. Define π∗ =argmaxπ [µπ −L (π)] and Ψ ∗ ≡ µπ∗−L (π∗). Note that we are charging all theidling penalty to a single supplier. In this way, a price π∗ guarantees a lower boundfor the payoffΨi(πi,π−i). Hence,Ψ ∗ is the payoff that any supplier can guarantee forhimself, i.e., his minmax level. We further let π :=Ψ ∗/µ = π∗−L (π∗)/µ < π∗ andobserve that choosing price π < π is a dominated strategy. Thus, π can be regardedas a lower bound of suppliers’ rational pricing strategies.

Although a standard approach is to look for a pure-strategy Nash equilibrium, inthe next proposition we show that none exists. Instead, we shall focus on the mixed-strategy competitive equilibrium. Let G(π) denote the mixing cumulative probabil-ity distribution of a supplier’s pricing strategy π . The next proposition characterizesthe structure of these mixing probabilities.

Proposition 3 With homogeneous rates, there exists a unique symmetric equilib-rium in which all suppliers randomize continuously over [π,π∗], and every suppliergets Ψ ∗. The randomizing distribution is G(π) = [ µ(π−π)

L (π) ]1/(n−1),∀π ∈ [π,π∗].

The reason for not having any pure-strategy equilibrium is intuitively due to thediscontinuity of suppliers’ revenue functions. This creates an incentive for the cheapsuppliers to increase their prices all the way to π; however, they would also avoidto reach π when themselves become the most expensive and incur a discontinu-ous penalty. Note that the range over which the price is randomized is completelydetermined by the individual’s problem. In all generic cases, no tie of the higheststatic price may occur. In other words, the market idleness process is contributedby only one supplier. Moreover, any tie of two prices takes place with probabilityzero, which is in contrast to the centralized system where prices are always equal.Therefore, our homogeneous service model suggests that price dispersion can beregarded as a sign of incoordination. Also note that in equilibrium, the expectedpayoff of a supplier is identical to the case where he carries the entire market idle-


ness, and hence he receives on average the minmax level. Competitive behaviordrives away the possibility of extracting additional revenues.

Having characterized the competitive equilibrium, we now turn to the market ef-ficiency issue. Define ΠC ≡ maxπ µπ−L (π) as the aggregate (second-order)revenue under the centralized solution and Π ∗ as the aggregate revenue among sup-pliers in the unique competitive equilibrium. The next proposition shows that theefficiency loss can be arbitrarily large when the number of suppliers explodes.

Proposition 4 Suppose that the service rates are homogeneous. For any given ag-gregate service rate µ , for any given constant M, there exists a sufficiently largenumber NM such that |ΠC−Π ∗|> M, ∀n > NM .

Proposition 4 shows that as the number of suppliers grows, the competitive be-havior among the suppliers may result in an unbounded efficiency loss. This demon-strates a significant inefficiency due to the market mechanism and it therefore callsfor the need of a coordination scheme, as we investigate in §4.5. Note that this state-ment is asymptotic in the sense of the number of suppliers, which is different fromthe case in §3, and it is particularly relevant in the context of the large-scale systemsdiscussed in §1. By restricting ourselves to the case of fixed aggregate service rate,we can then illustrate that the efficiency loss that results from the market idlenessterm also plays a pivotal role.

Note also that the first-order aggregate revenues of the centralized solution andthe competitive equilibrium coincide; nevertheless, this is by construction of theasymptotic regime specified in §3.

4.4.2 Heterogeneous service rate case

Now we consider the scenario where suppliers are endowed with different ser-vice rates. We again first define a global maximizers π∗1 , π∗2 ,..., π∗n , if supplieri (1 ≤ i ≤ n) is the one who proposes the highest price solely; i.e., we defineπ∗i = argmaxπi [µiπi−L (πi)] and Ψ ∗i ≡ µiπ

∗i −L (π∗i ) as the global maximum rev-

enue that supplier i can achieve as J = i. Next we let π i =Ψ ∗i /µi and recall thatchoosing price π < π i is a dominated strategy for supplier i. The following proposi-tion characterizes the relevant properties of an equilibrium needed for our purpose.Gi(·) denotes the mixing distribution that supplier i adopts in equilibrium.

Proposition 5 Suppose suppliers are endowed with heterogeneous service rates.Then in a competitive equilibrium,

• All Gi(·)’s have the same left endpoint (denoted by s) of their supports. Moreover,s≥maxi∈N π i.

• Suppliers’ expected payoffs are proportional to their service rates µi’s.• If n = 2 and µ1 > µ2, then there exists a unique equilibrium in which supplier i’s

revenue is µiπ1, i = 1,2. The equilibrium mixing probabilities are respectively


G2(π) =µ1(π−π1)

L (π),∀π ∈ [π1,π

∗1 ], G1(π) =

µ2(π−π1)

L (π),∀π ∈ [π1,π

∗1 ), (31)

and G1(π∗1 ) = 1. G1(π) = G2(π) = 0,∀π ≤ π1.

The first result on left endpoints is not surprising. This comes directly from ananalogous argument for Proposition 3. The second result captures the ex ante differ-ence between suppliers’ payoff function: higher service rate brings higher equilib-rium payoff. When we restrict to the duopoly setting, we know perfectly the rangeover which suppliers randomize their prices, and we can obtain closed-form expres-sions for their expected payoffs. They randomize the prices over the same range,and the supplier with a higher service rate tends to set a lower price: his mixingdistribution stochastically dominates the other’s in the usual, first-order sense. Thisimplies that when a supplier has a capacity advantage, he can afford to price lowerto capture more customers.

4.4.3 Numerical results

In this section, our goal is to compare the performance between the centralizedsolution and the competitive equilibrium. We consider a system with n M/M/1servers, delay sensitivity parameter c = 0.5, and arrival rate of buyers Λ = 1.The valuation v of each customer is assumed to follow an exponential distribu-tion with mean 0.1, and p is set such that the effective arrival rate P(v ≥ p)matches the total service rate µ . As an example, if we let µ = e−1.3, then p canbe obtained as follows: Λe−10p = µ ⇔ p ≈ 0.13. The other relevant parameter isγ = f (p)/(1−F(p)) = 10. Note that as we scale according to Λ = r and µ = µr, pstays unchanged.

The next two figures compare the centralized solution and the competitive equi-librium. Take n = 2 and assume µ = µ1 + µ2 = e−1.3. Without loss of general-ity, we assume that supplier 1 has a higher capacity and let a ≡ µ1

µ∈ (0.5,1)

denote the heterogeneity of service rates between these two suppliers. Figure 3demonstrates the mixing distributions of supplier 1 under a competitive equilib-rium with different values of a. Figure 4 presents the upper and lower bounds ofthe price for the mixing distributions. Note that the mixing distribution may havea point mass at the upper bound π∗1 , in which case the mixing distribution jumpsto 1 at π∗1 (e.g., a = 0.57,0.64,0.71 in Figure 3). Although the mixing distribu-tions of supplier 2 have no point mass, the comparison of the mixing distributionsacross different degrees of heterogeneity is qualitatively similar and therefore isomitted. Combining Figure 3 and Figure 4, there is no unambiguous prediction forthe suppliers’ pricing decisions when the capacity heterogeneity increases. The in-crease of the heterogeneity, a, has two effects. First, it mitigates the competitionbetween the suppliers due to the difference of capacities. This might induce higherprices. Second, the increase of a also increases the variance of the service time(since σ = (( 1

aµ)2 +( 1

(1−a)µ )2)1/2 is increasing in a ∈ (0.5,1)). This increases the


magnitude of the second-order price through the parameter β . Since the second-order price is negative, it implies that the suppliers would set a lower price whenthe variance is higher. Because of these two conflicting forces, no clear rankingof the mixing distribution can be obtained (as seen in Figure 3), and the boundsare not monotonic (in the same direction) as the degree of heterogeneity increases(see Figure 4). Note also that in the centralized solution, only one price is set:

πC ≡ argmaxπ

µπ− pγ µβ

φ(π/β )

1−Φ(π/β )

∈ [4.5,6.0] when a∈ [0.5,0.71]. Since

πC is strictly positive, the prices in the competitive equilibrium are significantlylower than the price under the centralized control.

−12 −11 −10 −9 −8 −7 −6 −5 −4 −30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

π

Mix

ing

dist

ribut

ion

of p

rice

G(⋅); a=0.5G

1(⋅); a=0.57

G1(⋅); a=0.64

G1(⋅); a=0.71

Fig. 3 The mixing distribution of prices versusthe heterogeneity of service rates.

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85−18

−16

−14

−12

−10

−8

−6

−4

−2

0

a (heterogeneity)

pric

e π

upper boundlower bound

Fig. 4 The bounds of prices versus the hetero-geneity of service rates.

Finally, we investigate how the number of suppliers affects the efficiency gap be-tween the centralized solution and the competitive equilibrium. To this end, we fo-cus on the case with homogeneous suppliers. This allows us to fully characterize theequilibrium pricing strategies and the suppliers’ expected (second-order) revenues.We first assume µ = e−1.3 and increase n, the number of suppliers. The individualservice rate is µi = µ/n, ∀i ∈N . As demonstrated in Figure 5, the range of pricesbecomes more negative when more suppliers participate in the market, due to amore severe competition among suppliers. In Figure 6, we draw the expected aggre-gate (second-order) revenue of the market, ∑i∈N Ψi(πi,π−i), and vary the numberof suppliers. We find that the expected aggregate revenue decreases when there aremore suppliers due to the increasing price competition (as presented in Figure 6).Thus, the mis-coordination problem becomes more serious when more suppliersparticipate in the market.


−40 −35 −30 −25 −20 −15 −10 −50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

π

G(π

)n=2n=3n=4

Fig. 5 The mixing distribution of prices versusthe number of suppliers.

1 2 3 4 5 6−30

−25

−20

−15

−10

−5

0

n

Exp

ecte

d ag

greg

ate

prof

it

Fig. 6 The expected aggregate revenue versusthe number of suppliers.

4.4.4 A remark on the suppliers’ participation

In characterizing the equilibrium behavior of the suppliers’ pricing game, we haveneglected the suppliers’ participation decisions. Note that the pricing decisionsπ j’s only affect the suppliers’ second-order revenues, which are simply smallperturbations around the first-order revenues pµi. (as seen in Lemma 1). Thus, asupplier is willing to participate if and only if his first-order revenue pµi is posi-tive, which depends on the capacity (service rate) decisions rather than the pricingdecisions.

To study the capacity game, we can assume that each supplier incurs a cost ofcapacity, ci(µi). Since the pricing decisions do not affect the first-order term, thesuppliers choose their capacities to maximize

maxµi≥0

pµi− ci(µi), (32)

where p is endogenously determined through F(p)=∑ j∈N µ j, i.e., p= F−1(∑ j∈N µ j).The function F−1(∑ j∈N µ j) can be interpreted as the inverse demand function,since it represents the customers’ effective arrival rate given the aggregate capacity∑ j∈N µ j. Notably, the above capacity game does not involve any stochastic term.

Moreover, according to [25, Corollary 1], this capacity game has a unique equi-librium if the following conditions are satisfied: 1) µF−1(µ) is concave in µ; 2)ci(µi) is weakly convex in µi; and 3) there exists a sufficiently large µ∗ such thatµF−1(µ)− ci(µ) is decreasing in µ when µ > µ∗. The first condition is related tothe price elasticity of the demand, the second condition implies a diseconomy ofscale for the capacity investment, and the third condition simply ensures that theaggregate market payoff never explodes. These conditions are widely adopted inmany surplus sharing games, which contains the celebrated Cournot competition as


a special case, to ensure that the competitive equilibrium is well-behaved (see [25]and the references therein).

Finally, a supplier is willing to participate in the market whenever in equilibriummaxµi≥0 pµi− ci(µi) ≥ 0. If we consider a special case in which the marginal costof capacity is constant, i.e., ci(µi) = ciµi, ∀µi, it is verifiable that only the suppliersthat are more cost efficient will participate, ie., the ones for whom ci < p.

4.5 Coordination scheme

The above competitive equilibrium analysis reveals that each supplier receives anexpected payoff lower than what he would obtain under the centralized solution.This implies that the centralized solution Pareto dominates all decentralized equi-libria. Thus, implementing a coordination scheme results in no conflict of interests,even though the suppliers may be ex ante heterogeneous with respect to servicerates. In addition, as the market size grows, the competitive behavior among sup-pliers may result in an unbounded efficiency loss. This demonstrates a significantinefficiency due to the market mechanism and motivates the search for a coordina-tion scheme.9

4.5.1 Sufficient condition for coordination

We will first study the suppliers’ behavior if they were “forced” to share the penalty,or revenue loss, that arises due to the market idleness. Under this scheme, supplieri’s payoff is

ΨPS

i (πi,π−i)≡ µiπi−µi

µL (max

j∈Nπ j), (33)

where the superscript PS refers to penalty sharing according to the service rates.The first term µiπi is the gross revenue supplier i earns by serving customers, andthe second term

µi

µL (max

j∈Nπ j) corresponds to his penalty share that is proportional

to his service rate µi. Note that this scheme is budget-balanced, i.e., no financingfrom outside parties is required. Let

πPS

i′ s denote the equilibrium prices under

this sharing scheme. Under this sharing scheme, the centralized solution can beachieved.

Proposition 6 Under the penalty sharing schemes that satisfy (33), πPSi = πC,∀i∈

N is the unique equilibrium.

9 It is worth mentioning that the mixed-strategy equilibrium is studied mainly to demonstrate thediscrepancy between the centralized system and the decentralized market equilibrium. It is con-ceivable that the implementation or identification of such a mixed-strategy equilibrium requiresfairly sophisticated communication and consensus among the suppliers. Nevertheless, the Paretodominance result justifies why such a coordination scheme is required and desired irrespective ofthe implementation issue of the mixed-strategy equilibrium.


Under the PS scheme, a supplier’s objective is in fact an affine function of theaggregate revenue (29). Hence, this coordination mechanism eliminates the wrongincentives of suppliers, regardless of the number of suppliers and their service rates.Since in both the competitive and the coordinated equilibria the suppliers’ expectedpayoffs are proportional to their service rates, all suppliers have a natural incentiveto join.

4.5.2 “Compensation-while-idling” mechanism that achieves coordination

The natural question is whether we can implement a penalty-sharing mechanismbased on observable quantities. We now show that this is achievable through anappropriate set of transfer prices between suppliers when one or more suppliers areidling.

Let ηi j be the transfer price per unit of time from supplier i to supplier j whensupplier j is idle in the limit model, with ηii = 0. The second-order revenue processfor a supplier i under this compensation scheme becomes

rPSi (t) = ri(t)+ δi(t), (34)

where

ri(t) ≡ µiπit + pσiBs,i(t)−µi pYi(t), (35)

δi(t) ≡ ∑j∈N , j 6=i

η jiYi(t)− ∑j∈N , j 6=i

ηi jYj(t). (36)

According to (35), the three terms correspond to his revenue from serving cus-tomers. In (36), δi(t) corresponds to the net transfers for supplier i : ∑ j∈N , j 6=i η jiYi(t)is the compensation he receives from other suppliers during the idle period, and∑ j∈N , j 6=i ηi jYj(t) is the cash outflow to other suppliers while compensating theiridleness.

Given (34), supplier i’s long-run average revenue can be expressed as

Ψi(πi,π−i) ≡ limt→∞

1t[ri(t)+ δi(t)] (37)

= µiπi−µi pE[Yi(∞)]+ ∑j∈N , j 6=i

η jiE[Yi(∞)]− ∑j∈N , j 6=i

ηi jE[Yj(∞)].

The next proposition specifies a set of transfer prices that implement the PSscheme.

Proposition 7 The transfer prices

ηi j =µiµ j

µp,∀i 6= j, i, j ∈N , and ηii = 0,∀i ∈N , (38)

implement the PS rule, i.e., Ψi(πi,π−i) =Ψ PSi (πi,π−i).


The transfer prices proposed in Proposition 7 essentially eliminate the imbalancebetween the current share of the market idleness incurred by an individual supplierand his required share (

µi

µL (max

j∈Nπ j)). Given these transfer prices, every supplier’s

objective is aligned with the centralized system (i.e., the objective is Ψ PSi (πi,π−i) in

(33)), and thus all suppliers are induced to set prices equal to πC.Proposition 7 shows that we are able to achieve coordination since given any

chosen prices, we can align the suppliers’ objectives with the planer’s objective. Toimplement this compensation scheme in the original system, we can simply requesteach supplier make transfers according to (38). This mechanism can be implementedand monitored by the intermediary in the market.

Note that the coordination scheme is independent of the static prices πi’s; itonly requires the information of the service rates µi : 1≤ i≤ n, which are publiclyavailable in our model. In fact, to facilitate the coordination scheme, the marketintermediary needs to have access to the current queue lengths, and should be ableto perfectly observe the idleness of suppliers.

4.6 Simulation results

To close the loop, we shall return to the original system described in §3.2 and seehow the competitive equilibrium and coordination scheme fare. To this end, weagain run simulations using the Arena model illustrated in Figure A1 of §3.6. Theparameters are the same as those in §4.4.3: c = 0.5, Λ = 1, µ = e−1.3, and thevaluation v is exponentially distributed with mean 0.1.

Mixing distributions of prices. Compared with §3.6, a new challenge arises:we cannot arbitrarily assign prices, because now the suppliers determine their com-petitive prices as equilibrium outcomes. We shall use the results from §4.4.3 asinputs to our Arena model for both homogeneous and heterogeneous cases of sup-pliers. We note that the equilibrium pricing strategy is described by a continuousdistribution without simple expressions (see Propositions 3 and 5). Thus, the usualinverse-transform method fails to apply (because the inverse of distribution functionis not known). Furthermore, the acceptance-rejection method is also not suitable forthis problem, because it requires an explicit expression of density function that isnot available. Our treatment follows from a similar idea to Figure 3. We first dis-cretize them and record the cumulative distribution at discrete points. We choosethe mesh sufficiently small and make linear interpolation to replicate approximatelythe original continuous distribution.

Second-order revenues. In terms of suppliers’ profits, we focus exclusively onthe pricing game in which the second-order correction around Rr

i (t) defined as

rri (t) ≡

1√r(Rr

i (t)− r pµit),∀i ∈ N . Note that given µ = e−1.3, p ≈ 0.13. When

there are two suppliers (n = 2), we let a≡ µ1

µ∈ (0.5,1) denote the heterogeneity of


service rates between these two suppliers. We examine two scenarios: in the homo-geneous case a = 0.5, these two suppliers are endowed with the same service rate(capacity). In the heterogeneous case, we choose a = 0.71.

Regarding the tie-breaking rule, here we examine two rules: the smallest indexfirst rule by which supplier 1 gets the priority, and the random priority rule by whichcustomers choose between tied suppliers with equal probabilities. For the randompriority rule, we can revise the conditions inside the ”Decide Suppliers” modulein Figure A1. Specifically, we add a ”two-way by chance” module to rout the cus-tomers randomly with 50-50 chances when there is a tie. For all the following sim-ulations, we conduct 1,000 replications, each of which takes the warm up of 120arrivals and regular simulation of 600 arrivals in expectation. The scaling factor r isfixed at 1,000. The confidence level is set at 5% when we make the statistical state-ments of hypothesis testing. We use two-sample-t two-tailed tests when we compareacross different scenarios and paired-t tests when comparing between the two sup-pliers within each scenario.

Revenue comparison: Symmetric case. First, we consider two symmetric sup-pliers (a= 0.5) and compare the suppliers’ second-order revenues in the competitiveequilibrium and under the coordination scheme. We find that the average differenceof suppliers 1’s and 2’s revenues in these two scenarios are statistically significant.For supplier 1, the estimated revenue difference is 0.758 whereas the 95% confi-dence interval is 0±0.00419. Similarly, supplier 2’s estimated revenue difference0.76 and 0.76 falls outside ±0.00421. Therefore, the coordination scheme indeedleads to higher expected revenues for both suppliers.

Revenue comparison: Asymmetric case. Second, we consider two asymmet-ric suppliers (a = 0.71). In this case, the coordination scheme again yields higherexpected revenues for both suppliers that are statistically significant. The estimatedrevenue improvements for suppliers 1 and 2 are 0.992 and 0.351 respectively, andthey fall outside the 95% confidence intervals ±0.0026 and ±0.00284. We can alsocompare the two suppliers’ revenues. Naturally, their (second-order) revenues aredifferent due to heterogeneous service rates. Using paired-t tests, we observe thatthe revenue differences are statistically significant in both the competitive and coor-dinated scenarios (-0.435 and 0.206 on average, and their corresponding confidenceintervals are ±0.00295 and ±0.00488.

Tie-breaking rule. Third, we can also examine the impact of tie-breaking rule.For this matter, we use the symmetric supplier case as illustration. We first start withthe competitive equilibrium and compare the two tie-breaking rules: smallest indexfirst and random rules. For the competitive equilibrium we fail to reject the nullhypothesis, i.e., the suppliers’ revenues are statistically indistinguishable under thetwo rules. In contrast, under the coordination scheme the tie-breaking rule matterssubstantially. The estimated revenue difference is 0.0857 and it falls outside the 95%confidence interval ±0.00323.

The above discrepancy can be explained intuitively. In the competitive equilib-rium, both suppliers randomize their prices. Thus, the chance of seeing an actual tieis infinitesimal (zero probability in theory). Therefore, the tie-breaking rule rarelycomes in action. However, under the coordination scheme, both suppliers are in-


duced to set prices at πC. Because their service rates are identical, often times tiesactually happen and the priority rule goes in favor of supplier 1. In this case, randomtie-breaking ensures the fair routing between suppliers and it leads to statisticallysignificant consequences. Further to the above observation, we run additional com-parisons between the two suppliers. Under the smallest index first rule, supplier 1earns on average 0.176 more than supplier 2, which is outside the 95% confidenceinterval ±0.00308. Under the random priority rule, this difference is negligible (-0.000792 on average).

To summarize, our simulations suggest that (1) the coordination scheme is effec-tive in both homogeneous and heterogeneous scenarios, and this benefit applies toall suppliers; (2) tie-breaking rules are inconsequential when suppliers adopt ran-domized pricing, but they do matter when instead deterministic prices are chosen.

5 Conclusions

We study an oligopolistic model in which suppliers compete for buyers that areboth price and delay sensitive. We apply both fluid and diffusion approximationsto simplify the multi-dimensional characteristics of the decoupled suppliers into asingle-dimensional aggregated problem. Specifically, we establish the “state spacecollapse” result in this system: the multi-dimensional queue length processes at thesuppliers can be captured by a single-dimensional workload process of the aggre-gate supply in the market, which can be expressed explicitly as a reflected Ornstein-Unlenbeck process with analytical expressions. Based on this aggregated workloadprocess, we derive the suppliers’ long-run average revenues and show that the sup-pliers’ competition results in a price randomization over bounded ranges, whereasunder the centralized control suppliers should set identical and deterministic prices.

To eliminate the inefficiency due to the competition, we propose a novel compensation-while-idling mechanism that coordinates the system: each supplier gets monetarytransfers from other suppliers during his idle periods. This mechanism alters sup-pliers’ objectives and implements the centralized solution at their own will. Theimplementation only requires a set of static transfer prices that are independent ofthe suppliers’ prices and the queueing dynamics such as the current queue lengthsor the cumulative idleness. Its simplicity is an appealing feature to be considered forpractical implementations in intermediary platforms such as online exchanges.

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Design of an aggregated marketplace under congestioneffects: Asymptotic analysis and equilibrium

characterizationAPPENDIX

Ying-Ju Chen∗ Costis Maglaras † Gustavo Vulcano‡

Notation. We use the following notation throughout the paper. We say that f (x) =o(g(x)) when f (x)/g(x)→ 0 when x→ ∞, and that f (x) = g(x) to state the factthat f (x) = g(x)+ o(

√x) or f (x) = g(x)+ o(1/

√x). Similarly, f (x) = op(g(x)) if

f (x)/g(x)→ 0 in probability. We say that f (x) = O(g(x)) if f (x)/g(x)→ a, fora constant a > 0 and define f (x) = Op(g(x)) as the counterpart of convergence inprobability. For any integer k > 0, and for any y∈ Rk, we define |y| := max|y j|, j =1, ...,k as its maximum norm. For any function f : R+ → Rk and constant L > 0,we define || f (·)||L := sup 0≤t≤L| f (t)|. For any sequence ar,r ∈ N, ar → α if forany δ > 0, there exists rδ such that |α − ar| < δ , whenever r > rδ . The functionφ(·) denotes the standard normal density, and Φ(·) its corresponding cumulativedistribution function (c.d.f.).

A1 Schematic descriptions of the simulation model

We run simulations using Arena and the model is illustrated in Figure A1, where weassign the (randomly generated) valuation as an ”attribute” to each customer. In themodule ”Decide Suppliers”, we compare the customer’s valuation and the full price(i.e., the price plus the expected waiting time) of each supplier. This determineswhether a customer should balk (when the valuation is too low), and when not balk-ing, which supplier to go to. Regarding the tie-breaking rule, we use the smallestindex first rule: when a customer feels indifferent between suppliers 1 and 2, shejoins supplier 1 by default. Other modules are self-explanatory. Arena software al-lows us to record the frequencies our specified variable (such as system workload)falls into distinct intervals. Thus, we can record the proportion of time the systemworkload falls below the boundary and above.

∗ School of Business and Management & School of Engineering, The Hong Kong University ofScience and Technology, Clear Water Bay, Kowloon, Hong Kong; tel: +852-2358-7758; fax: +852-2358-2421; email: [email protected]† Columbia Business School, 409 Uris Hall, 3022 Broadway, New York, NY 10027,[email protected]‡ School of Business at Universidad Torcuato di Tella, Buenos Aires, Argentina, [email protected]


Fig. A1 The simulation model in Arena.

A2 Proofs of main results

Proof of Proposition 1

Define κi(t) = πi +cQi(t)

µi,∀i ∈N as the total cost supplier i submits for the buyer

at epoch t, and κ(t) = [κ1(t), ...,κn(t)]. Let J(t)⊆N be the set of suppliers tied asthe cheapest at time t and Jc(t) ≡N \ J(t) be its complement. In this continuousfluid limit model, we can imagine that buyers arrive in a continuous manner andare routed to the cheapest supplier(s) accordingly. Given that this routing process iscontinuous, oscillations, that would be a core feature of the discrete buyer setting,will never occur. Critically, the buyers in our fluid limit model are not counted oneby one; rather, they are infinitesimally small.We hereby formalize the above statement. Define the function

g(κ(t)) = maxi, j∈N

(κi(t)−κ j(t)) = maxi∈N

κi(t)− minj∈N

κ j(t). (A1)

Note that g(κ(t)) is nonnegative, and g(κ(t)) = 0 if and only if κi(t) = κ j(t), ∀i, j ∈N . When g(κ(t)) > 0, the arrivals will not be routed to the most expensive sup-plier at time t, and the cheapest supplier accumulates customers. We will show thatg(κ(t)) can be used as a Lyapunov function to prove that |κi(t)− κ j(t)| → 0 ast→ ∞, and subsequently conclude the statement of the proposition.Note that as long as not all suppliers receive new orders, the aggregate arrival rate,that equals ∑i∈N µi, is always higher than the aggregate service rate of servers thattie as the cheapest ones. That is, ∑i∈N µi > ∑ j∈J(t) µ j for all t before the resourcepooling regime (if it exists). Let k ∈ J(t) be an arbitrary cheapest supplier. We nowclaim that for any time epoch, the following equality holds:

κk(t) =cµk

µk

∑i∈J(t) µi

[∑

i∈Nµi− ∑

i∈J(t)

µ j

], (A2)


where ∑i∈N µi−∑i∈J(t) µ j is the imbalance between the aggregate arrival rate and

aggregate service rate for those cheapest suppliers, andµk

∑i∈J(t) µiis the proportion

of customers routed to supplier k; the term cµk

simply accounts for the sensitivity ofκk(t) on the queue length increase.We prove Equation (A2) by contradiction. Suppose, on the contrary, that (A2) isviolated at some time for some supplier in the cheapest suppliers’ set. Let

t1 := inft|t, κk(t) 6=cµk

µk

∑i∈J(t) µi

[∑

i∈Nµi− ∑

i∈J(t)

µ j

], f or some k ∈ J(t) (A3)

be the time right after which violation occurs and (with abuse of notation) letk be such a supplier. There are two cases for this violation: Case 1) κk(t1) <c

µk

µk∑ j∈J(t1)

µ j

[∑i∈N µi−∑ j∈J(t1) µ j

]and Case 2) κk(t1)> c

µk

µk∑ j∈J(t1)

µ j


].

Consider the first case. Recall J(t1) is the set of suppliers tied as the cheapest at timet1 and define ε1 := mini∈Jc(t1) κi(t1)−min j∈J(t1) κ j(t1) as the mimimum differencebetween those κ j(t1)’s inside and outside J(t1) at time t1. If ε1 = 0, the problem isnon-existent and we have already reached the resource pooling regime. Thus, belowwe consider the case ε1 > 0.Because mini∈Jc(t1) κi(t1)−min j∈J(t1) κ j(t1)> 0, the inequality κk(t1)< c

µk

µk∑ j∈J(t1)

µ j


]is strict, and κk(t)’s are all continuous functions, we can find a sufficiently smallδ < ε1

c(1+∑i∈N µi/min j∈N µ j)such that 1) the sets J(t)⊆ J(t1), for all t ∈ [t1,t1+δ ), and

2) κk(t)< cµk

µk∑i∈J(t) µi

[∑i∈N µi−∑i∈J(t) µ j

], for all k ∈ J(t), for all t ∈ [t1,t1 +δ ).

Consider any i1 ∈ Jc(t1) and j1 ∈ J(t1). For t ∈ [t1,t1 +δ ), we note that

κi1(t)−κ j1(t)

= πi1 + cQi1(t)

µi1−

π j1

+ cQ j1

(t)

µ j1

≥ πi1 + cQi1(t)−δ µi1

µi1−

π j1

+ cQ j1

(t)+δ ∑i∈N µi

µ j1

≥ πi1 + cQi1(t)

µi1−

π j1

+ cQ j1

(t)

µ j1

− cδ

1+

∑i∈N µi

µ j1

≥ ε1− cε1

c(1+∑i∈N µi/min j∈N µ j)

1+

∑i∈N µi

µ j1

> 0.

Therefore, within the time window [t1, t1 +δ ), no supplier in Jc(t1) will join the setof cheapest suppliers. This suggests that all the arrivals will be routed to J(t1) withinthe time window [t1, t1 +δ ) and J(t)⊆ J(t1) for t ∈ [t1,t1 +δ ).


Now we focus on the set J(t1). Because κk(t1)< cµk

µk∑ j∈J(t1)

µ j

[∑i∈N µi−∑ j∈ J (t1)

µ j

],

there must exist another supplier l in J(t1) such that κl(t)> cµl

µl∑ j∈J(t1)

µ j


].

To see this, recall that the aggregate arrival rate is ∑i∈N µi during [t1,t1 +δ ), whichis always higher than ∑ j∈J(t1) µ j, the aggregate service rate of servers that tie asthe cheapest ones in J(t). Since no supplier in Jc(t1) receives any arrival during[t1, t1 + δ ), the queue length accumulation rate is at least ∑i∈N µi−∑ j∈J(t1) µ j (ifJ(t) is strictly smaller than J(t1), then the queue length accumulation rate can onlygo higher). If for all l ∈ J(t1)\k, κl(t)≤ c

µl

µl∑ j∈J(t1)

µ j


], then

the queue length accumulation of each l is

Ql(t)≤µl

c

cµl

µl

∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

]=

µl

∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

].

Likewise, for supplier k we have Qk(t) <µk

∑ j∈J(t1)µ j


]. Thus,

the aggregate queue length accumulation among all suppliers is

Qk(t)+ ∑l∈J(t1)\k

Ql(t)+ ∑i∈Jc(t1)

Qi(t)

<µk

∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

]+ ∑

l∈J(t1)\k

µl

∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

]+0

< ∑i∈N

µi− ∑j∈J(t1)

µ j,

which leads to a contradiction.Given that κl(t)> c

µl

µl∑ j∈J(t1)

µ j


], we derive the corresponding

κl(t) as follows (l ∈ J(t1)\k):

κl(t) > κl(t1)+cµl

µl

∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

](t− t1)

= πl + cQl(t1)

µl+

cµl

µl

∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

](t− t1)

= πk + cQk(t1)

µk+

c∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

](t− t1),

where the last equality comes from the fact that both suppliers k and l belong toJ(t1). On the other hand, for supplier k we have:


κk(t) < κk(t1)+cµk

µk

∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

](t− t1)

= πk + cQk(t1)

µk+

c∑ j∈J(t1) µ j

[∑

i∈Nµi− ∑

j∈J(t1)

µ j

](t− t1).

Thus, κl(t)> κk(t), for all t ∈ [t1,t1 +δ ). However, this implies that supplier l shallnever receive any arrival during [t1,t1+δ ), and its queue length can only drop during[t1,t1 +δ ). This leads to a contradiction.The second case, κk(t) > c

µk

µk∑ j∈J(t1)

µ j


], is the mirror image

of the above. In this case, we must be able to find another supplier l ∈ J(t1) suchthat κl(t)< c

µl

µl∑ j∈J(t1)

µ j


]. Otherwise, the queue length accu-

mulations will be violated within the set J(t1). However, we can then observe thatκk(t)> κl(t) in a small time window and supplier k shall never receive any arrival.This implies that κk in that small time window is strictly negative and leads to acontradiction. Collectively, Equation (A2) must hold.Next, we return to the proposed Lyapunov function g(κ(t)). We shall prove thatthere exists a lower bound for the depreciation rate of g(κ(t)). We observe thatddt (maxi∈N κi(t)) is non-positive, because those suppliers in Jc(t) do not receivearrivals and their queue lengths can only depreciate. On the other hand, Equation

(A2) shows thatddt

(minj∈N

κ j(t))≥ 0, because ∑i∈N µi−∑i∈J(t) µ j > 0 and there-

fore the derivative is always non-negative. Thus, g(κ(t)) = ddt (maxi∈N κi(t))−

ddt

(min j∈N κ j(t)

)is strictly negative and |g(κ(t))| > |κk(t)| = κk(t), where k ∈

J(t). We have

|g(κ(t))| ≥ κk(t)=cµk

µk

∑i∈J(t) µi

[∑

i∈Nµi− ∑

i∈J(t)

µ j

]≥ c

∑i∈N µi−min j∈N µ j

[∑

i∈Nµi− ∑

i∈J(t)

µ j

].

(A4)From ∑i∈N µi − ∑i∈J(t) µ j ≥ min j∈N µ j as long as J(t) 6= N , we then have

|g(κ(t))| ≥cmin j∈N µ j

∑i∈N µi−min j∈N µ j. Note that this lower bound is independent of

epoch t and is strictly positive; the tie-breaking rule does not matter due to the con-tinuity of queue length processes. To wit, our argument to establish Equation (A2)does not hinge on any specific tie-breaking rule. Thus, the above result applies toall tie-breaking rules. When J(t) =N , the system has reached the resource poolingregime.

Let 4κ be such that πl + cQl(0)

µl+4κ = max

i∈Nπi + c

Qi(0)µi. In words, 4κ

is the amount of cost that supplier l has to add so that his κl(0) ties the maxi-mum initial cost. Because the imbalance of proposed costs is decreasing in t ata guaranteed rate, the κi(t)’s will become equal before time s∗(Q(0),W (0)) =


4κ(∑i∈N µi−min j∈N µ j)

cmin j∈N µ j. Thus, g(κ(t)) = 0, ∀t ≥ s∗. The next step needed is

to bound 4κ as a function of M0. Call it 4(M0). Then for any δ > 0, s∗(δ ,M0) ≤

s∗(M0) =4(M0)(∑i∈N µi−min j∈N µ j)

cmin j∈N µ j. This completes the proof for the conver-

gence of fluid model. ut


The proof builds on the framework advanced by [8]. To facilitate exposition wewill follow [8] very closely, stating and addressing the differences and the requiredmodifications.A brief sketch of the proof is as follows. Step 1. Hydrodynamic limits: LemmasA1-A5 will establish that appropriately scaled processes that focus on the systembehavior over order 1/

√r time intervals satisfy the fluid equations (11)-(16). Step 2.

Convergence of diffusion scaled processes: Combining the above with Proposition1, we will establish that the supplier queue length process is close to the “balanced”state configuration for all time t; i.e., the short transient digressions are not visiblein the natural time scale of the system.Step 1:The hydrodynamic scaling we adopt here is the following (cf. [8, Equation (5.3)]):For m = 1,2, ...,b

√rτc,

Qr,m(t) =1√r

Qr(1√r

t +1√r

m),

Ar,m(t) =1√r[Ar(

1√r

t +1√r

m)−Ar(1√r

m)],

Dr,m(t) =1√r[Dr(

1√r

t +1√r

m)−Dr(1√r

m)],

T r,m(t) = T r(1√r

t +1√r

m)−T r(1√r

m),

Y r,m(t) = Y r(1√r

t +1√r

m)−Y r(1√r

m),

X r,m(·) = (Qr,m(·),Ar,m(·),Dr,m(·),T r,m(·),Y r,m(·)),

where Qr,Ar,Dr,T r, and Y r are all n-dimensional vectors. Note that for cumulativeprocesses Ar(·),Dr(·),T r(·), and Y r(·), the associated hydrodynamic scale processesAr,m(t),Dr,m(t), and T r,m(t),Y r,m(t), account for the cumulative changes between

[1√r

m,1√r

t +1√r

m]. On the contrary, Qr,m(t) only keeps record of the values at


epoch1√r

t+1√r

m. The scaling 1√r is used to ensure that the scaled processes admit

meaningful limits as r→ ∞.We now analyze the processes introduced above and provide probabilistic bounds.This essentially is equivalent to [8, Proposition 5.1]. For ease of notation, let usdefine Qr,m(t) = Qr,m

i (t). The routing indicator is Ii(Qr,m(t)) = 1 if ΞJ( 1√r t+ 1√

r m) =

i, and 0 otherwise. Note that√

rQr,mi (t) = Qr

i (1√r

t +1√r

m) is the queue length of

supplier i at epoch1√r

t +1√r

m. Therefore,

p+πi√

r+ c√

rQr,mi (t)+1rµi

= p+πi√

r+ c

Qri (

1√r t + 1√

r m)+1

µri

(A5)

is simply the total cost submitted by supplier i sinceQr

i (1√r t + 1√

r m)+1

µri

is the delay

quotation. We further define

Jri (Q

r,mi (t)) = P

v≥ p+

πi√r+ c√

rQr,mi (t)+1rµi

,∀i ∈N (A6)

as the probability that at time epoch1√r

t +1√r

m, the valuation of a new arrival, v,

exceeds the total cost submitted by supplier i. The following lemma establishes theprobability bounds.

Lemma A1 Fix ε > 0, L > 0, and τ > 0. For r large enough,

P

maxm<√

rτ

maxi∈N||Ar,m

i (t)−∫ t

0Λ

rIi(Qr,m(u))Jri (Q

r,mi (u))du||L > ε

≤ ε, (A7)

P

maxm<√

rτ

maxi∈N||Dr,m

i (t)−µiTr,m

i (t)||L > ε

≤ ε, (A8)

P

max

m<√

rτ

maxi, j∈N

||(

πi + cQr,m

i (t)µi

)−

(π j + c

Qr,mj (t)

µ j

)||L > ε

≤ ε. (A9)

The next lemma shows that the hydrodynamic processes are “nearly Lipschitz con-tinuous.”

Lemma A2 Fix ε > 0, L > 0, and τ > 0. There exists a constant N0 such that for rlarge enough,

P

max

m<√

rτ

supt1,t2∈[0,L]

|X r,m(t2)−X r,m(t1)|> N0|t2− t1|+ ε

≤ ε. (A10)


The above two lemmas imply that the measure of “ill-behaved” events is negligiblefor the hydrodynamic scaled processes. Note that (A9) only shows the convergencein probability for one particular instance. Nevertheless, the ultimate state space col-lapse requires an aggregation/ integration over all instances, and continuous integra-tions over measure-zero events could lead to non-trivial consequences.Let N0 denote the constant required in Lemma A2 and focus on the complementof these ill-behaved events. We can choose a sequence ε(r) decreasing to 0 suffi-ciently slowly such that the inequalities (A9), (A8), and (A10) still hold. For such asequence ε(r), we define

Kr0 =

max

m<√

rτ

supt1,t2∈[0,L]

|X r,m(t2)−X r,m(t1)| ≤ N0|t2− t1|+ ε(r)

,

Kr1 =

maxm<

√rτ maxi∈N ||Ar,m

i (t)−∫ t

0 Λ rIi(Qr,m(u))Jri (Q

r,mi (u))du||L ≤ ε(r);

maxm<√

rτ maxi∈N ||Dr,mi (t)−µiT

r,mi (t)||L ≤ ε(r)

maxm<√

rτ maxi, j∈N ||(

πi + c Qr,mi (t)µi

)−(

π j + cQr,m

j (t)µ j

)||L ≤ ε(r);

,

Kr = Kr0 ∩Kr

1.

Note that the set Kr contains those events that possess our desired properties. Thenext step is to show that when r is sufficiently large, we can restrict our attentionto these well-behaved events. This is parallel to [8, Proposition 5.2, Corollary 5.1].and follows from Lemmas A1 and A2.

Lemma A3 P(Kr)→ 1, as r→ ∞.

Let E be the space of functions x : [0,L]→ R5 that are right continuous and have leftlimits. Define E

′= x ∈ E : |x(0)|< M0, |x(t2)−x(t1)|< N0|t2− t1|,∀t1, t2 ∈ [0,L],

where M0 is a fixed constant. Let Er0 = X r,m(·,ω),m <

√rτ,ω ∈ Kr denote the

sets of well-behaved events and E0 = Er0,r ∈ R+ is the collection of these sets.

We next show that the set of candidate hydrodynamic limits is “dense” in the statespace: when r is sufficiently large, the vector of processes X r,m(·,ω) is close to somecluster point of E0. A function X is said to be a cluster point of the functional spaceE0 if for all δ > 0, there exists a Xδ ∈ E0 such that ||X−Xδ ||L < δ .

Lemma A4 Fix ε > 0,L > 0, and τ > 0. There exists a sufficiently large r(ε) suchthat for all r > r(ε), for all ω ∈ Kr and for all m = 1,2, ...,b

√rτc, ||X r,m(·,ω)−

X(·)||L < ε, for some X(·) ∈ E0∩E ′ with X(·) being a cluster point of E0.

The above lemma follows closely from [8, Proposition 6.1]. Given that all hydrody-namic scale processes have a close-by cluster point, it remains to study the behaviorof these cluster points. The next step proves that all these cluster points are in factsolutions to the deterministic fluid equations (11)-(16) (cf. [8, Proposition 6.2]).

Lemma A5 Fix L > 0 and τ > 0. Let X(·) be an arbitrary cluster point of E0 over[0,L]. Then X(·) satisfies the fluid equations (11)-(15).


Step 2:Combining the above results with Proposition 1, we will now establish the desiredstate space collapse property for the supplier queue length processes. Let us first fixconstants η ,ξ ,ε > 0. By Lemma A3, there exists a sufficiently large r(ξ )> 0 suchthat P(Kr)> 1−ξ for all r > r(ξ ).Take L = s(ε,M0) + 1 where s(·, ·) was defined in Proposition 1. Let r(η) be

sufficiently large such thatL√r< η whenever r > r(η). Now we consider the

diffusive scaled processes in the time interval [ L√r ,τ]. For all ς ∈ [

L√r,τ], let

mr(ς) = minm ∈ N+ : m <√

rς < m + L = maxd√

rς − Le,0 and τ′r(ς) :=

√rς −mr(ς). Thus, ς =

1√r

[τ′r(ς)+mr(ς)

]. Straight-forward algebra shows that

Qri (ς) =

1√r

Qri (ς) =

1√r

Qri

(1√r

τ′r(ς)+

1√r

mr(ς)

)= Qr,mr(ς)

i (τ′r(ς)). From the

definition of τ′r(ς), if r > r(η), for all ς ∈ [

L√r,τ], we have

τ′r(ς)=

√rς−mr(ς)=

√rς−maxd

√rς−Le,0≥

√rς−(

√rς−L−1)=L−1= s(ε,M0).

(A11)This implies that the convergence of fluid scale process can be applied at time ς ∈[

L√r,τ] when r > r(η). Moreover, mr(ς)≤

√rς ≤

√rτ by construction. Therefore,

the convergence of hydrodynamic scale processes is valid here for all ς ∈ [L√r,τ]

as well.

We now verify that the imbalance between πi + cQr

i (ς)

µiand π j + c

Qrj(ς)

µ jis upper

bounded (note that the additional terms 1/µri ’s in the suppliers’ bids have been

taken into account in Lemma A1 through (A9)):

maxi, j∈N

|(

πi + cQr

i (ς)

µi

)−

(π j + c

Qrj(ς)

µ j

)|

≤ maxi, j∈N

|(

πi + c Qr,mr(ς)i (τ

′r (ς))

µi

)−(

πi + c Qi(τ′r (ς))

µi

)|+ |

(πi + c Qi(τ

′r (ς))

µi

)−(

π j + c Q j(τ′r (ς))

µ j

)|

+|(


µ j

)−(

π j + cQr,mr(ς)

j (τ′r (ς))

µ j

)|

≤ 2max

i∈N|

(πi + c

Qr,mr(ς)i (τ

′r(ς))

µi

)−

(πi + c

Qi(τ′r(ς))

µi

)|

+ maxi, j∈N

|

(πi + c

Qi(τ′r(ς))

µi

)−

(π j + c

Q j(τ′r(ς))

µ j

)|.


From Lemmas A1 and A4, for fixed L,τ,ξ , η , and ε > 0, there exists a sufficientlylarge r(L,τ,ξ , η ,ε)>maxr(ξ ),r(η) such that for all r > r(L,τ,ξ , η ,ε),ω ∈Kr,for all ς ∈ [ L√

r ,τ], we have

maxi∈N|

(πi + c

Qr,mr(ς)i (τ

′r(ς))

µi

)−

(πi + c

Qi(τ′r(ς))

µi

)| ≤ ε

3, (A12)

and maxi, j∈N |

(πi + c

Qi(τ′r(ς))

µi

)−(


µ j

)| ≤ ε

3 . This implies that

maxi, j∈N |(

πi + cQr

i (ς)

µi

)−(

π j + cQr

j(ς)

µ j

)| ≤ ε

3 +ε

3 +ε

3 = ε. Note that P(Kr)→ 1

as r→∞, i.e., the state space collapse holds in probability (the additional term 1/µri

is of order 1/r and therefore does not affect the result as demonstrated in LemmaA1). We can then choose appropriate L,τ,ξ , η , and ε such that the proposition isestablished. ut

Proof of Theorem 1

We start with a sketch of the proof, and then provide the details for each part.Step 1. Write down the equation of W r(t). Recall that the workload process is

W r(t) = ∑i∈NQr

i (t)µr

i, where Qr

i (t) = Qri (0) + Ar

i (t)− Sri (T

ri (t)), and the scaled

workload process W r(t) =√

rW r(t). We first apply strong approximations ([14,Theorem 5]) on the cumulative arrival process routed to each supplier and the ser-vice requirement processes, and then the state space collapse result to derive the ex-pression of W r(t) as a function of the total arrival process Ar(t). Note that potentialbuyers that find it costly to purchase and choose not to purchase are automaticallyleft out from the effective arrival process.Step 2. Fluid model properties of cumulative idleness and aggregate market demand.Let Y r

i (t) = t − T ri (t) denote the market idleness, where T r

i (t) is the cumulativework completion for supplier i up to time t, and Λ r(t) be the aggregate arrival rateinto the market. We will prove that Y r

i (t) → 0, in probability, u.o.c.,∀i ∈N , andΛ r(t)

r→ ( ∑

i∈Nµi)t, u.o.c.

Step 3. Use strong approximations and oscillation inequalities to bound U r(t) andW r(t). In this step, we approximate W r(t) using the reflection maps (ϕ,ψ) (similarto the framework in [26]) and apply Lemma 7 in [5] to bound the market idlenessU r(t) and workload W r(t).Step 4. Establish the weak convergence of Zr(t). Having established the conver-gence for individual idleness process, we now focus on the process W r(t). We willfollow [20] and use Gronwall’s inequality to bound the difference of W r(t) from an-


other process that has the “desirable” limit, and then apply convergence togetherlemma to establish the weak convergence. The weak convergence of Qr(t) fol-lows immediately from the state space collapse result and the convergence togetherlemma.

Complete Proof of Theorem 1

Step 1:

From Proposition 2, we have πi + cQr

i (t)µi

= π + cW r(t)+ op(1), ∀i ∈ N . In the

sequel we can simply focus on those events in which state space collapse arises.Recall that the queue length process can be rewritten as Qr

i (t) = Qri (0)+Ar

i (t)−Sr

i (t−Y ri (t)), where Sr

i (·) is the cumulative service completions when the underlyingrandom service times are i.i.d. with mean 1

rµiand standard deviation σi

r . Then,

πi + cQr

i (t)µi

= π + cW r(t)+op(1),

⇒ πi + cQr

i (0)µi

+ cAr

i (t)√rµi− cSr

i (t−Y ri (t))√

rµi= π + cW r(t)+op(1).

Rearranging the above equation, we see that

Ari (t)√

r=

µi

c[π + cW r(t)]+

Sri (t−Y r

i (t))√r

− µiπi

c− Qr

i (0)+op(1), (A13)

and therefore the aggregate market demand Ar(t)≡ ∑i Ari (t) satisfies

Ar(t)√r

=∑i∈N µi

c[π+ cW r(t)]+ ∑

i∈N

Sri (t−Y r

i (t))√r

−∑i∈N µiπi

c− ∑

i∈NQr

i (0)+op(1).

(A14)Next we focus on the demand and service time processes. A strong approximation([14, Theorem 5]) for the Poisson process N(·) allows us to rewrite Ar(t)=N(Λ r(t))as

Ar(t) = Λr(t)+Ba(Λ

r(t))+O(logrt), (A15)

where Λ r(t) =∫ t

0 λ r(mini∈N [p+ πi√r +c

√rQr

i (s)+1rµi

])ds is the arrival rate of aggregatemarket demand, and Ba(·) is a standard Brownian motion and the subscript “a”stands for “arrival.”Similarly, according to [12], we can apply strong approximation to service comple-tion process t−Y r

i (t) (which is a random time) and represent the renewal processSr

i (·) as

Sri (t−Y r

i (t)) = µri t−µ

ri Y r

i (t)+√

rσiBs,i(t−Y ri (t))+O(logr[t−Y r

i (t)]), (A16)


where Bs,i(·) is the standard Brownian motion associated with supplier i’s servicetime distribution, and the subscript “s” stands for “service.” 1 Note that Ba(t) andBs,i(t)’s describe distinct Brownian motion processes and they are mutually inde-pendent. The service rate is of order r, and therefore the Brownian motion is scaledby√

r.

From Proposition 2 we have that mini∈N

πi + c

Qri (t)µi

+ c1µr

i

= π + cW r(t)+

op(1), ∀t, and P(W r(t)√

r> 0)→ 0, uniformly in t. Therefore, the arrival rate can be

expressed as

λr(

mini∈N

p+

πi√r+ c√

rQri (s)+1rµi

)= λ

r(

p+π + cW r(s)√

r+op(

1√r)

)= λ

r(p)− λ r(p)′

√r

[π + cW r(s)]+op(√

r) = ∑i∈N

µri −√

r( ∑i∈N

µi)f (p)F(p)

[π + cW r(s)]+op(√

r),

where the second equality follows from a first-order Taylor expansion at p = p andthe last equality follows from the fact that p induces full resource utilization and thedefinition of λ r(·). With this expression, the cumulative rate of aggregate marketdemand Λ r(t) becomes

Λr(t) = ( ∑

i∈Nµ

ri )t−

√r( ∑

i∈Nµi)

f (p)F(p)

∫ t

0[π + cW r(s)]ds+op(

√rt). (A18)

Combining (A14), (A15), (A16), and (A18), and for γ ≡ f (p)F(p)

, we obtain

1√r

( ∑

i∈Nµ

ri )t− ( ∑

i∈Nµi)γ

∫ t

0[π + cW r(s)]ds

+

Ba(Λr(t))√r

+op(1)

=∑i∈N µi

c[π + cW r(t)]+

∑i∈N µri√

rt− ∑i∈N µr

i√r

Y ri (t)+ ∑

i∈NσiBs,i(t−Y r

i (t))−∑i∈N µiπi

c− ∑

i∈NQr

i (0).

We can rearrange the equation as follows:

∑i∈N µi

c[π + cW r(t)] =−( ∑

i∈Nµi)γ

∫ t

0[π + cW r(s)]ds+

√r ∑

i∈NµiY r

i (t)+∑i∈N µiπi

c+ ∑

i∈NQr

i (0)

1 More specifically, Corollary 5.5 of Chapter 7 in [12] ensures the existence of a probability spacein which a Poisson process N and a Brownian motion B exist such that

supt≥0

|N(t)− t−B(t)|log(2∨ t)

< ∞, a.s. (A17)

See also [20, §7.2, p. 268] for a similar treatment.


+Ba(Λ

r(t))√r

− ∑i∈N

σiBs,i(t−Y ri (t))+op(1).

Recall that Zr(t) = π + cW r(t), µ ≡ ∑i∈N µi, and Y ri (t) =

√rY r

i (t),∀i ∈ N .

From the assumption of initial queue lengths πi + cQr

i (0)µi

= π + cW r(0), ∀i ∈N ,

we obtain Qri (0) =

µi

c(Zr(0)− πi), and therefore that ∑i∈N Qr

i (0) =µ

cZr(0)−

∑i∈N µiπi

c.

Using these substitutions, we get that

µ

cZr(t)=−µγ

∫ t

0Zr(s)ds+ ∑

i∈NµiY r

i (t)+Z(0)+[Ba(Λ

r(t))√r

− ∑i∈N

σiBs,i(t−Y ri (t))]+op(1).

(A19)Defining U r(t) =

cµ

∑i∈N

µiYi(t) as the “total market idleness,” we conclude that

Zr(t)= Z(0)−γc∫ t

0Zr(s)ds+U r(t)+

cµ

[Ba(Λ

r(t))√r

− ∑i∈N

σiBs,i(t−Y ri (t))

]+op(1).

(A20)Step 2:In this section, we will establish the convergence of scaled copies of Λ r(t) and Y r

i (t)according to

Λ r(t)r→ µt, inprobability, u.o.c. (A21)

Y ri (t)→ 0, inprobability, u.o.c.,∀i ∈N . (A22)

First we prove (A21). From (A18), we haveΛ r(t)

r= µt− µγ√

r

∫ t

0Zr(s)ds+op(

1√r),

where the last term vanishes in the limit. From Proposition 2, we know that for any

fixed constant C, P(supt≤T W r(t) > C)→ 0, and hence P(supt≤TW r(t)√

r> ε)→

0, ∀ε > 0. Moreover,µγ√

r

∫ t

0Zr(s)ds=

µγ√r[πt+c

∫ t

0W r(s)ds]→ 0, in probability, u.o.c.,

as r→ ∞, because the first term inside the brackets is a constant and the integral is

uniformly bounded by ε√

r. Therefore,Λ r(t)

r→ µt, inprobability, u.o.c., which

completes the proof of (A21).Next we show (A22) by contradiction. Suppose that there exists a supplier i such thatY r

i (t)0 u.o.c., then we can find a constant ε1 > 0 and a sequence (rk, tk) such thatrk→∞ as k→ ∞ and Y rk

i (tk)> ε1, ∀k. Along this sequence, due to the nonnegativityof Y r

j (t)’s and the scaling√

r for Y ri (t), we obtain U rk(tk) >

√rk(cµiε1/µ). Thus,

as k→ ∞, U rk(tk)→ ∞. From (A20), this implies that Zrk(tk)→ ∞, and thereforethat W rk(tk)→ ∞. This, however, contradicts the fact that for all T > 0, ε > 0, there


exists a constant C such that P(supt≤T˜W r(t)>C)< ε. This completes the proof of

(A22).Step 3:We can rewrite (A20) as Zr(t) = Zr(0)− γc

∫ t0 Zr(s)ds + U r(t) + V r(t), where

V r(t) =cµ[Ba(Λ

r(t))√r

− ∑i∈N

σiBs,i(t−Y ri (t))]. Recalling that

Λ r(t)r→ µt a.e. and

Y ri (t)→ 0 in probability, we can apply the invariance principle of Brownian motion

and convergence together lemma to obtain

Ba(Λr(t))√r

D=Ba(Λ r(t)

r)⇒Ba(µt)D=

√µBa(t), ∑

i∈NσiBs,i(t−Y r

i (t))⇒ ∑i∈N

σiBs,i(t)D=σBs(t),

(A23)

where σ ≡√

∑i∈N σ2i and Ba,Bs are independent standard Brownian motions.

Thus, V r(t)⇒ cµ[√

µBa(t)−σBs(t)]D=cµ

√σ2 + µB(t), where B(t) is a standard

Brownian motion and the second expression follows again from the invariance prin-ciple. Since Y r

i (t) is nonnegative, non-decreasing, and continuous, we have U r(t) iscontinuous, non-decreasing, and U r(t)≥ 0.Now we quote a technical lemma:

Lemma A6 Let εr be a real-valued sequence such that εr→ 0 as r→ ∞, and ζ ≡1c maxi (πi− π). Then,

∫ t0 11|W r(s)−ζ |> εrdU r(s) → 0 in probability, u.o.c.

Motivated by the result in Lemma A6, we will rewrite U r(t) in two parts as follows:

U r(t) =∫ t

011|W r(s)−ζ |> ε

rdU r(s)︸︷︷︸Ur

ε (t)

+∫ t

011|W r(s)−ζ | ≤ ε

rdU r(s)︸︷︷︸Ur

ζ(t)

.

(A24)Note that U r

ζ(·),U r

ε (·) are nonnegative, continuous, and nondecreasing. Moreover,U r

ε (t)→ 0, as r→ ∞ in probability, u.o.c., and supt≤T |U rζ(t)−U r(t)| → 0 in prob-

ability. Define now the process V (t) =cµ

√σ2 + µB(t), and consider the auxiliary

process R(t) defined as follows: R(t) = R(0)− γc∫ t

0 R(s)ds+V (t)+U(t), whereU(t) is continuous and nondecreasing, U(0) = 0, and U(t) increases only whenW (t) hits ζ , i.e., only when R(t) = π + cζ = maxi∈N πi ≡ π . Note also that π canbe regarded as the lower reflecting barrier of the limiting process of R(t). We lateron show that this coincides with the limit of U r(t). By construction, R(t) has thebehavior of the hypothesized limit for Zr(t) specified in Theorem 1.Let X(t) = R(t)− π . Then,

X(t) = X(0)− γc∫ t

0[X(s)+ π]ds+V (t)+U(t)≡ ϕ(X(t)), (A25)

where ϕ(·) is the reflection operator ([20]) and X(t) is defined by X(t) = X(0)−γc∫ t

0 [X(s)+ π]ds+V (t).


The remaining goal is to show that Zr(t) converges to R(t). Recall that Zr(t) =Zr(0)−γc

∫ t0 Zr(s)ds+V r(t)+U r

ζ(t)+U r

ε (t), Zr(t)= Zr(0)−γc∫ t

0 Zr(s)ds+V r(t)+

U rζ(t)+U r

ε (t), and define Hr(t) = Zr(t)− π and its primitive process Hr(t) as fol-lows: Hr(t) = Hr(0)− γc

∫ t0 [H

r(s)+ π]ds+V r(t), and Hr(t) = Hr(t)+U r(t).The key remaining element of the proof is to show that Hr(s)⇒ X(t) in distribu-tion, which will imply the weak convergence of Zr(t) to R(t). From Lemma A6and Proposition 2, for any sequence εr > 0, s.t. εr → 0 as r → ∞, there exists asequence δ r > 0 where δ r ↓ 0 and r large enough such that for all r > r, we haveP(infs≤t W r(s)≥ ζ − εr) = 1−δ r. Let Ω(εr) be the set of sample paths for whichinfs≤t W r(s)≥ ζ − εr. Note that P(Ω(εr)) = 1−δ r→ 1 as r→ ∞.In the sequel we will use Lemma 7 of [5]. We first observe that ∀ω ∈Ω(εr), W r(t)≥ζ − εr⇔ Zr(t)≥ π− cεr⇔ Hr(t)≥−cεr, and, similarly, W r(s)> ζ + εr impliesHr(s) > cεr. If we define Hr

1(t) = Hr(t)+ cεr, Hr2(t) = Hr(t)+ cεr, and focus on

the event ω ∈Ω(εr), then Hr1,H

r2,U

r satisfy the following conditions:

Hr1(t)=Hr

2(t)+U r(t),Hr1(t)≥ 0,U r(·) nondecreasing; U r(0)= 0,

∫ t

011Hr

1(s)> 2cεrdU r(s)= 0,

(A26)where the last equation follows from a simple transformation Hr(s) > cεr ⇔Hr

1(s) > 2cεr. Since Hr, Hr,U r all are right continuous and have left limits in[0,T ], ∀T > 0, these processes Hr

1,Hr2,U

r fit the conditions of Lemma 7 in [5].Thus, if we define ψ(X(t)) = supX(s)− : 0≤ s≤ t where X− = max(0,−X), andϕ(X) ≡ X −ψ(X) is the regulated process of X , we obtain from Lemma 7 of [5]that ψ(Hr

2(t))≤ U r(t)≤ ψ(Hr2(t))+2cεr.

Recall that ψ(·) is nonincreasing and Lipschitz continuous with unity constant, andHr

2(t) = Hr(t)+ cεr. It follows that

ψ(Hr(t))− cεr ≤ U r(t)≤ ψ(Hr(t))+2cε

r, (A27)

which in turn implies ϕ(Hr(t))−2cεr ≤ Hr(t)≤ ϕ(Hr(t))+ cεr.Step 4:Subtracting Hr(t) by the auxiliary process X(t), we obtain

Hr(t)− X(t) =−γc∫ t

0[Hr(s)− X(s)]ds+V r(t)−V (t)+op(1),

⇒ |Hr(t)− X(t)| ≤ γc∫ t

0|Hr(s)− X(s)|ds+ |V r(t)−V (t)+op(1)|,

where the last inequality follows from the triangle inequality. Recall that ϕ(·) isLipschitz continuous and let K denote the Lipschitz constant. Thus, using (A27), thisinequality can be rewritten as |Hr(t)− X(t)| ≤ γcK

∫ t0 |Hr(s)− X(s)|ds+ |V r(t)−

V (t)+op(1)|+2γc2εrt.From the strong approximation for V r(t), we have that for any sequence vr > 0,vr ↓0, there exists r large enough such that


supt≤T|V r(t)−V (t)| ≤ vr, w.p. 1−θ

r, (A28)

where θ r ↓ 0 as r→ ∞. In the sequel we concentrate on sample paths in Ω(εr) forwhich (A28) holds, i.e., we consider only ω ∈ Ω(εr,vr) ≡ Ω(εr)∩Ω(vr), whereΩ(vr) is the set of sample paths for which supt≤T |V r(t)−V (t)| ≤ vr. The measureof Ω(εr,vr) is more than 1−δ r−θ r. Then,

|Hr(t)− X(t)| ≤ γcK∫ t

0|Hr(s)− X(s)|ds+ vr +2γc2

εrt ≤ (vr +2γc2

εrT )eγcKT ,

(A29)where the second inequality follows from Gronwall’s inequality and the fact thatt ≤ T .From the Lipschitz continuity of ϕ(·), (A25), and (A29), we get that |Hr(t)−X(t)| ≤ K(vr +2γc2εrT )eγcKT +2cεr. Let r→∞, sequences εr,vr,δ r,θ r all vanishand therefore we conclude that

sup t≤T |Hr(t)− X(t)| → 0 inprobability. (A30)

From (A30), the fact that Zr(t) = Hr(t)+ π , and the convergence together theorem,it follows that Zr(t) converges to R(t), which is the desired result identified in The-orem 1. The weak convergence of the queue length processes follows immediatelyfrom the convergence of Zr(t) and the state space collapse result of Proposition 2.ut


We will follow [6] to prove this result. Let s and s denote respectively the essentialupper and lower bounds of G(·)’s support. That is, G(π) = 0,∀π < s,0 ≤ G(π) <1, ∀π ∈ (s, s), and G(π) = 1,∀π ≥ s. If the support is not bounded above, we cansimply set s = ∞. We further let α(π) denote the mass probability a supplier putson price π .First we observe that s ≥ π, ∀i ∈ N . To see this, if a supplier chooses π∗, hisexpected payoff is at least µπ∗ −L (π∗) = Ψ ∗ = µπ. Thus, choosing any priceπ < π yields an expected payoff no more than µπ < µπ = Ψ ∗, and therefore isstrictly dominated. Moreover, α(s) = 0, i.e., no supplier is expecting to be the mostexpensive while placing s. We verify the latter by contradiction: Suppose α(s) >0, and that a supplier selects s. He will become the most expensive supplier (andtherefore shares the penalty) when all other suppliers also select s, which occurs withprobability [G(s)]n−1. When this occurs, all n suppliers share the penalty equally,and therefore the supplier’s expected payoff is

µs− 1nL (s)[G(s)]n−1 < µs. (A31)


Since G(s) is strictly positive, we are subtracting a positive amount in the LHS. Nowsuppose that a supplier deviates and chooses a price s− ε while other suppliers setprices at s. Thus, his expected payoff is µ(s− ε) (without any penalty incurred).When ε is small enough, µ(s− ε) is strictly higher than the LHS of (A31). Thisimplies that the supplier will always intend to undercut the price by a sufficientlysmall ε .We now define Ψi(π) as the expected payoff of supplier i if he places price π , and Ψ e

ihis equilibrium payoff. We claim that the suppliers get equal payoffs in equilibrium,i.e., Ψ e

i = Ψ ej = µs, ∀i, j ∈ N . The proof is as follows. Under a mixed-strategy

equilibrium, a supplier should get the same payoff from all strategies on which heplaces positive probabilities. Hence Ψ e

i =Ψi(s) = µs, ∀i ∈N .Next, we characterize the support. Suppose that α(s) > 0. Then transferring theweight α(s) to a price just below it will be a profitable deviation. Therefore α(s) =0. Given that, when a supplier places price s, with probability one he will be thesole most expensive supplier and hence carries the entire market idleness. Thus, ifs 6= π∗, his payoff would be

Ψi(s) = µ s−L (s)< µπ∗−L (π∗) = µπ ≤ µs, (A32)

where the strict inequality follows from that π∗ is the unique maximizer of µπ −L (π). This leads to a contradiction, since Ψ e

i = µs, ∀i ∈N . Thus, s = π∗. Finally,since

µs =Ψe

i =Ψi(π∗) = µπ

∗−L (π∗) = µπ, (A33)

we conclude that s = π .Having characterized the support, we now show that the suppliers will randomizecontinuously over the entire support. We first claim that there is no point mass in(π,π∗). Suppose this is not true. Then there exists a price π ∈ (π,π∗) such that eachsupplier puts a point mass on π . Following the argument in Lemma 10 of [6], itpays for a supplier to transfer a ε-neighborhood mass above π to a δ -neighborhoodbelow π , and thus this cannot be an equilibrium.Now we show that G is strictly increasing in the entire support. If the claim is false,there must exist an interval (π1,π2) such that G(π1) = G(π2). Since [G(π)]n−1 (i.e.,the probability that a supplier quoting π becomes the most expensive supplier) doesnot change in (π1,π2), by moving a small mass from slightly below π1 to π2, asupplier is strictly better off. That is, the supplier charges a higher price π , with asmaller probability of becoming the most expensive. This contradicts the assump-tion that G(π) is an equilibrium strategy, and therefore G must be strictly increasing.Combining all above, in a symmetric equilibrium suppliers randomize continuouslyin the support. Finally, we derive the closed-form expression for G(π). Since G(π)is strictly increasing in the entire support, π is involved in supplier i’s equilibriumstrategy. Therefore a supplier should get the same payoff for all such π’s (otherwisehe would not be willing to play the equilibrium strategy). Thus, For all π ∈ [π,π∗],

Ψe

i = µs =Ψi(π) = µπ−L (π)[G(π)]n−1,∀π ∈ [π, π∗], (A34)


where the last equality follows from that with probability [G(π)]n−1 supplier i willbecome the most expensive one. Rearranging the above equation, we obtain theexpression for G(π).Finally, the above derivations also imply that no pure-strategy equilibrium exists,because the formulation actually allows for probability mass in G(π). As we verifyhere, there always exist profitable deviations in such a scenario. This completes theproof. ut


Given µ (and other relevant parameters such as c and σ ), L (π) is a known functionand therefore ΠC is a fixed constant independent of n, the number of suppliers.

From Proposition 3, Π ∗ = nΨ ∗, where Ψ ∗ = maxπ [µπ −L (π)] = maxπ [µ

nπ −

L (π)]. The function L (π) is strictly convex and positive, and therefore Ψ ∗ can beobtained via the first-order condition. When n is sufficiently large, the maximizer

π∗ = argmaxπ [µ

nπ −L (π)] is arbitrarily small. Thus, for a given number M, we

obtain that there exists a number such that π∗ < min−M−ΠC

µ,

ΠC

µ whenever

n > NM . When n > NM, the difference between the aggregate revenues under thecentralized and decentralized systems is then

|ΠC−Π∗|=Π

C−Π∗=Π

C−nmaxπ

[µ

nπ−L (π)]>Π

C−nµ

nπ∗=−µπ

∗+ΠC >M.

(A35)This completes the proof. ut


Let si and si denote respectively the essential lower and upper bounds of Gi(·)’ssupport, and αi(π) is the probability mass supplier i puts at point π . Let Ψi(π,G−i)be the expected payoff of supplier i when he plays π and other suppliers adopt themixing probabilities G−i. Let Ψ e

i be his expected payoff in equilibrium.Since quoting a price below π i is a dominated strategy for supplier i, we have si≥ π i.Moreover, the lower bounds of supports for Gi(·)’s must be equal. If this is not thecase, a supplier with the lowest si would refuse to put any positive weight on pricesbetween his lower bound and the highest lower bound max j∈N s j. Combining theabove, we know that si := s≥max j∈N π j,∀i ∈N .Now we claim that if a supplier plays s, with probability 1 he will not be the mostexpensive supplier. That is, there exist i, j such that αi(s) = α j(s) = 0. If this is not


true, then transferring some weight to a neighborhood just below s will be profitablefor some supplier.Since at least two suppliers put zero mass at s, and every supplier may place s inequilibrium, we have

Ψi(s,G−i) = µis−L (s)×0 = µis, (A36)

which results in Ψ ei = µis, ∀i ∈N .

We now focus on the two-supplier case. The following lemma provides some re-lations of π∗1 ,π

∗2 and π1,π2, which are needed for characterizing the equilibrium

mixing probabilities.

Lemma A7 If µ1 > µ2, then π∗1 > π∗2 and π1 > π2.

We now return to the proof of Proposition 5. Note that the upper bounds of thesupports for both G1 and G2 must be the same. We argue by contradiction: If thisis not true, then we have either s1 > s2, or s2 > s1. Consider the first case. Notethat when supplier 1 places a price π ∈ (s2, s1], he will carry the entire market idle-ness. If π∗1 < s1, then transferring some distribution weight in (s2, s1] downwards tomax(π∗1 , s2) is a profitable deviation for supplier 1. If π∗1 > s1 , then he would liketo transfer all the weight in (s2, s1] upwards to π∗1 . Thus s1 > s2 is impossible. Sim-ilarly, one can show that the other case never occurs as well. Therefore s1 = s2 = s.Given that there are only two suppliers, both suppliers do not put a point mass onthe common upper bound simultaneously, otherwise a mass transfer from s to itslower neighborhood will be profitable for either supplier. Thus, at most one supplierputs probability zero on s. Suppose α1(s) = 0. Then

Ψ2(s,G−2) = µ2s−L (s)≤Ψ∗

2 = µ2π2 < µ2π1 ≤ µ2s, (A37)

which contradicts the equilibrium condition. Hence, from the strict inequality inbetween the extremes, α1(s)> 0 and consequently α2(s) = 0. Moreover,

Ψ1(s,G−1) = µ1s−L (s)≤ µ1π∗1 −L (π∗1 ) = µ1π1 ≤ µis =Ψ1(s,G−1), (A38)

and the only chance for equalities to hold is that s = π∗1 and s = π1. This determinesthe common support.In the interior, no hole and no point mass should be placed for both suppliers, oth-erwise one can construct a profitable weight transfer. Thus, both G1,G2 increasecontinuously in [π1,π

∗1 ). Finally, since in equilibrium a supplier should obtain the

same expected payoff at any point in [π1,π∗1 ], we have

Ψ1(π,G−1) = µ1π−G2(π)L (π) = µ1π1 =Ψ1(π1,G−1),∀π ∈ [π1,π∗1 ],

⇒ G2(π) =µ1(π−π1)

L (π),∀π ∈ [π1,π

∗1 ].

Similarly,


Ψ2(π,G−2) = µ2π−G1(π)L (π) = µ2π1 =Ψ2(π1,G−2),∀π ∈ [π1,π∗1 ),

⇒ G1(π) =µ2(π−π1)

L (π),∀π ∈ [π1,π

∗1 ),

and α1(π∗1 ) = 1− µ2(π

∗1 −π1)

L (π). This completes the proof. ut


From the PS rule, we can define Π PSi (πi,π−i) = µiπi− µi

µL (max j∈N π j) = µi[πi−

1µ

L (maxj∈N

π j)] as the revenue for supplier i when other suppliers select π−i. Note

that supplier i always benefits from raising his price infinitesimally if it has notbeen the highest. Thus, in equilibrium all suppliers must submit the same price. The

derivative of Π PSi with respect to πi becomes

∂Π PSi (πi,π−i)

∂πi= µi[1−

1µ

L′(max

j∈Nπ j)11πi =

maxj∈N

π j].

Recall from Lemma 3 that πC = argmaxπµπ−L (π) and therefore 1− 1µL′(πC)=

0. Imposing symmetry and plugging π j = πC,∀ j∈N , we obtain∂Π PS

i (πi,π−i)

∂πi|π j=πC ,∀ j∈N =

µi

[1− 1

µL′(πC)

]= 0, where the second equality follows from the definition of

πC. Therefore, every supplier setting price πC is the symmetric equilibrium underthis sharing rule. The uniqueness follows from the strict convexity of L (·). ut


Given the transfer prices ηi j = µiµ j/(µ p), i, j ∈ N , supplier i’s second-orderrevenue becomes

rPSi (t) = µiπit + pσiBs,i(t)−µi pYi(t)+ ∑

j∈N , j 6=iη jiYi(t)− ∑

j∈N , j 6=iηi jYj(t)

= µiπit + pσiBs,i(t)− pµi

µ∑

j∈Nµ jYj(t)

= µiπit + pσiBs,i(t)−µi p1c

U(t),


where we recall that U(t) =cµ

∑j∈N

µ jYj(t). Thus, supplier i’s long-run average

second-order revenue is Ψi(πi,π−i) = µiπi−µipc E[U(∞)] = µiπi−

µi

µL (max

j∈Nπ j) =

ΨPS

i (πi,π−i). ut

Proof of Lemma 3

Since the second term in (29) is independent of the lower static prices πi, i /∈ J, at op-timality these πi′s should all be equal, i.e., πi = π, ∀i∈N . Thus, the problem re-

duces to a single-parameter maximization problem: maxπ

µπ− pγ µβ

φ(π/β )

1−Φ(π/β )

.

Recall thatφ(z)

1−Φ(z)is the hazard rate of standard normal distribution and hence it

is increasing and convex ([27, §5]).2 Thus, πC is well-defined and unique. ut

A3 Proofs of auxiliary lemmas in Sections 3 and 4

Proof of Lemma 1

Recall that using the strong approximation theorem ([14, Theorem 7]), the servicetime process of supplier i can be expressed as follows:

Sri (t) = µ

ri (t−Y r

i (t))+√

rσiBs,i(t−Y ri (t))+O(logrt), (A39)

where Bs,i is the associated Brownian motion with standard deviation σi, and Y ri

is the corresponding idleness process. Recall that Rri (t) = (p+

πi√r)Sr

i (t). Thus, asr→ ∞,

Rri (t)r

=1r(p+

πi√r)[rµi(t−Y r

i (t))+√

rσiBs,i(t−Y ri (t))+O(logrt)

]→ pµit,

(A40)after removing the lower-order terms. ut

Proof of Lemma 2

Recalling the expression Rri (t) = (p+ πi√

r )Sri (t) from Lemma 1, we have

2 We thank Ramandeep Randhawa for bringing this reference to our attention.


1√rRr

i (t)− r pµit

=1√r

(p+

πi√r)[µ

ri t−µ

ri Y r

i (t)+√

rσiBs,i(t−Y ri (t))+O(logrt)

]− r pµit

=

1√r√

r[µiπit + pσiBs,i(t−Y ri (t))−µi pY r

i (t)]

+σiπiBs,i(t−Y ri (t))−µiπiY r

i (t)+O(logrt).

From Qri (t) = Qr

i (0) + Ari (t)− Sr

i (t −Y ri (t)), the idleness process can be repre-

sented as Y ri (t) = t − [Sr

i ]−1 (Qr

i (0)+Ari (t)−Qr

i (t)) , where [Sri ]−1 is well-defined

since Sri (·) is monotonically increasing. Since Ar

i (t), Qri (t) both converge according

to Theorem 1, the convergence Y ri (t) =

√rY r

i (t) to the limiting process Yi(t) thenfollows from Theorem 1 and the convergence together theorem. Moreover, sinceY r

i (t)→ 0 as r→ ∞ from the proof of Theorem 1, we get that

rri (t) =

1√r(Rr

i (t)− r pµit)⇒ µiπi + pσiBs,i(t)−µi pYi(t), as r→ ∞. ut

Proof of Lemma A1

Our first goal is to show that the scaled arrival process Ar,mi (t) has an upward jump

if and only if Ii(Qr,m(t)) = 1 and p +πi√

r+ c√

rQr,mi (t)+1rµi

≤ v, where v is the

customer’s willingness to pay, where the routing indicator clearly specifies whichsupplier should get a new buyer if her valuation is higher than the full price.To this end, we shall consider two events

p+πi√

r+ c√

rQr,mi (t)+1rµi

≤p+π j√

r+ c

√rQr,m

j (t)+1

rµ j (A41)

and

p+πi√

r+ c√

rQr,mi (t)+1rµi

≤ v. (A42)

We recall the definition of Qr,m(t) and establish the following equivalence:

p+πi√

r+ c√

rQr,mi (t)+1rµi

≤p+π j√

r+ c

√rQr,m

j (t)+1

rµ j

⇔ p+πi√

r+ c

Qri (

1√r t + 1√

r m)+1

µri

≤p+π j√

r+ c

Qrj(

1√r t + 1√

r m)+1

µrj

,

where we recall Qr,mi (t) =

1√r

Qri (

1√r

t +1√r

m) andQr

i (1√r t + 1√

r m)+1

µri

is simply

the expected delay a buyer encounters when joining the queue i at epoch1√r

t +


1√r

m. Thus, (A43) implies that the total cost submitted by supplier i is less than that

by supplier j.

Similarly, if p+πi√

r+ c√

rQr,mi (t)+1rµi

≤ v, we obtain

p+πi√

r+ c

Qri (

1√r t + 1√

r m)+1

µri

= p+πi√

r+ c√

rQr,mi (t)+1rµi

≤ v. (A43)

Therefore, if p +πi√

r+ c√

rQr,mi (t)+1rµi

≤p +π j√

r+ c

√rQr,m

j (t)+1

rµi, ∀ j 6= i, and

p +πi√

r+ c√

rQr,mi (t)+1rµi

≤p +π j√

r+ c

√rQr,m

j (t)+1

rµi, ∀ j < i, and p +

πi√r+

c√

rQr,mi (t)+1rµi

≤ v, then Ari (

1√r

t +1√r

m), the arrival process routed to server i

at time epoch1√r

t +1√r

m, has an upward jump. But this implies that Ar,mi (t) =

1√r

[Ar

i (1√r

t +1√r

m)−Ari (

1√r

m)

]also increases since Ar

i (1√r

m) is unchanged.

Therefore, the scaled arrival process Ar,mi (t) has an upward jump if and only if

Ii(Qr,m(t)) = 1 and p+πi√

r+ c√

rQr,mi (t)+1rµi

≤ v. Thus, the variation associated

with Ar,mi (t) for a specific supplier i is upper bounded by the variation associated

with the counting process of the aggregate arrivals, and this bound applies uniformlyto all the suppliers i ∈N . Consequently, we obtain that

r maxi∈N||Ar,m

i (t)−∫ t

0Λ

rIi(Qr,m(u))Jri (Q

r,mi (u))du||L ≤ ||N(t)− t||Lr, (A44)

where N(t) represents the counting process of the arrivals and we have applied

0≤ r∫ t

0Λ

rIi(Qr,m(u))Jri (Q

r,mi (u))du≤ Lr,∀t ∈ [0,L]. (A45)

Note that in the above discussions, we have applied the tie-breaking rule that thebuyer chooses the supplier with the smallest index. This is inconsequential as it isalso the buyer’s best response and thus can be sustained as an equilibrium.We can now establish the probability bound based on the above inequality:

P(

maxm<√

rτ

maxi∈N||Ar,m

i (t)−∫ t

0Λ

rIi(Qr,m(u))Jri (Q

r,mi (u))du||L > ε

)

≤b√

rτc

∑m=1

P||Ar,m

i (t)−∫ t

0Λ

rIi(Qr,m(u))Jri (Q

r,mi (u))du||L > ε


≤b√

rτc

∑m=1

P

1r||N(t)− t||Lr > ε

≤b√

rτc

∑m=1

ε

L2r

≤ b√

rτc 1L2r

ε,

where in the third second inequality we have applied [8, Proposition 4.3].The proof for the departure process is similar to the above argument, and thereforewe only present the major steps here. Consider the term Dr,m

i (t)− µiTr,m

i (t). Fromthe definition of hydrodynamic scaling, we have

Dr,mi (t)−µiT

r,mi (t)

=1√r[Dr

i (1√r

t +1√r

m)−Dri (

1√r

m)]−µri

[T r

i (1√r

t +1√r

m)−T ri (

1√r

m)

]=

1√r

Sr

i (Tr

i (1√r

t +1√r

m))−Sri (T

ri (

1√r

m))−µri T r

i (1√r

t +1√r

m)+µri T r

i (1√r

m)

=

1√r

[Sr

i (Tr

i (1√r

t +1√r

m))−µri T r

i (1√r

t +1√r

m)]− [Sri (T

ri (

1√r

m))−µri T r

i (1√r

m)]

,

where Sri (·) is the service completion process. Therefore, The probability bound

P

maxm<√

rτ

maxi∈N||Dr,m

i (t)−µiTr,m

i (t)||L > ε

≤ P

(max

m<√

rτ

maxi∈N

|| 1√

r

[Sr

i (Tr

i (1√r t + 1√

r m))−µri T r

i (1√r t + 1√

r m)]||L

+|| 1√r [S

ri (T

ri (

1√r m))−µr

i T ri (

1√r m)]||L

> ε

)

≤b√

rτc

∑m=1

∑i∈N

P

1√r ||[Sr

i (Tr

i (1√r t + 1√

r m))−µri T r

i (1√r t + 1√

r m)]||L

+ 1√r ||[S

ri (T

ri (

1√r m))−µr

i T ri (

1√r m)]||L

> ε

≤ nb√

rτcC1ε

r,

where C1 is an appropriate constant independent of r.

Finally, we consider the imbalance of πi + cQr,m

i (t)µi

and π j + cQr,m

j (t)

µ j. The imbal-

ance can be expressed as

πi + cQr,m

i (t)µi−

(π j + c

Qr,mj (t)

µ j

)

= πi + c1√r Qr

i (1√r t + 1√

r m)

µi−

(π j+c

1√r Qr

j(1√r t + 1√

r m)

µ j

)


= πi +√

rcQr

i (1√r t + 1√

r m)+1

rµi−

(π j +√

rcQr

j(1√r t + 1√

r m)+1

rµ j

)− c√

r(

1µi− 1

µ j)

=√

r

(p+

πi√r+ c

Qri (

1√r t + 1√

r m)+1

rµi

)−√

r

(p+

π j√r+ c

Qrj(

1√r t + 1√

r m)+1

rµ j

)

+c√r(

1µi− 1

µ j).

Therefore,

||πi + cQr,m

i (t)µi−

(π j + c

Qr,mj (t)

µ j

)||L

≤√

r||

(p+

πi√r+ c

Qri (

1√r t + 1√

r m)+1

rµi

)−

(p+

π j√r+ c

Qrj(

1√r t + 1√

r m)+1

rµ j

)||L

+c√r| 1µi− 1

µ j|.

Note that(

p+ πi√r + c

Qri (

1√r t+ 1√

r m)+1

rµi

)−(

p+ π j√r + c

Qrj(

1√r t+ 1√

r m)+1

rµ j

)is simply the

imbalance of the total cost submitted by suppliers i and j. The probability bound canthen be obtained as follows:

P

max

m<√

rτ

maxi, j∈N

||(

πi + cQr,m

i (t)µi

)−

(π j + c

Qr,mj (t)

µ j

)||L > ε

≤ P

maxm<√

rτ

√r maxi, j∈N ||

(p+ πi√

r + cQr

i (1√r t+ 1√

r m)+1

rµi

)−(

p+ π j√r + c

Qrj(

1√r t+ 1√

r m)+1

rµ j

)||L

> ε− c√r |

1µi− 1

µ j|

≤b√

rτc

∑m=1

P

maxi, j∈N ||(

p+ πi√r + c

Qri (

1√r t+ 1√

r m)+1

rµi

)−(

p+ π j√r + c

Qrj(

1√r t+ 1√

r m)+1

rµ j

)||L

> 1√r

(ε− c√

r |1µi− 1

µ j|)

≤b√

rτc

∑m=1

nP

1r||N(t)− t||Lr >

1√r

(ε− c√

r| 1µi− 1

µ j|)

≤ C2b√

rτc√

rr

(ε− c√

r| 1µi− 1

µ j|),

where the third inequality follows from a similar argument to bound the routingprocess, and C2 is an appropriate constant that is independent of r. This bound isarbitrarily small when r is sufficiently large and ε is small. This completes the proof.ut


Proof of Lemma A2

The proof is similar to that of [8, Proposition 5.2], and therefore we only discuss themain steps. Consider first Ar,m(t). Without loss of generality, let us assume t2 > t1.We have that

Ar,mi (t2)−Ar,m

i (t1) =1√r[Ar

i (1√r

t2 +1√r

m)−Ari (

1√r

m)]− 1√r[Ar

i (1√r

t1 +1√r

m)−Ari (

1√r

m)]

=1√r[Ar

i (1√r

t2 +1√r

m)−Ari (

1√r

t1 +1√r

m)],

and therefore

Ar,mi (t2)−Ar,m

i (t1)=1√r

N

(∫ 1√r t2+ 1√

r m

1√r t1+ 1√

r mΛ

rIi(Qr(t))P(v≥ minj∈N

p+

π j√r+ c

Qrj(t +m)+1

µrj

)dt

),

(A46)where N(·) is the counting process of a unit-rate Poisson. Note that the arrival rate is

bounded Λ rIi(Qr(t))P(v ≥ min j∈N

p+

π j√r+ c

Qrj(t +m)+1

µrj

) ≤ r. Therefore,

the mean of Ar,mi (t2)−Ar,m

i (t1) is1√r× r× (

1√r

t2−1√r

t1) = t2− t1. This implies

that Ar,mi (t) is nearly Lipschitz continuous with unit constant. Similarly, we can also

show that the scaled processes Dr,mi (t), i∈N are also nearly Lipschitz continuous

with Lipschitz constants µi’s. This parallels [8, Proposition 5.2] and hence thedetails are omitted.We now consider Qr,m

i (t), i ∈ N . Recall that the queueing dynamics Qri (t) =

Qri (0)+Ar

i (t)−Dri (t),∀i ∈N , we obtain that

|Qr,mi (t2)−Qr,m

i (t1)| = |1√r[Qr

i (1√r

t2 +1√r

m)−Qri (

1√r

t1 +1√r

m)]|

≤ 1√r|Ar

i (1√r

t2)−Ari (

1√r

t1)|+1√r|Dr

i (1√r

t2)−Dri (

1√r

t1)|.

From the above discussions on Ari and Dr

i , Qr,mi (t), i∈N are also nearly Lipschitz

continuous with constant maxi∈N µi+1. The Lipschitz continuity of T r,mi (t) and

W r,mi (t) can be established similarly. ut

Proof of Lemma A4

Although the routing in our model depends on the queue length processes Qr,mi (t), i∈

N , the Lipschitz continuity of X r,mi from Lemma A3 and the definition of Kr lead

to this lemma immediately according to [8, Proposition 4.1]. ut


Proof of Lemma A5

This follows from [8, Proposition 6.2]. The idea is to approximate the cluster pointX(·) by a sequence of X r,m(·), and then take r to the infinity. Specifically, let usconsider, for example, the queue length dynamics:

Qi(t) = Qi(0)+∫ t

0ΛIi(Q(u))du− Di(t),∀i ∈N . (A47)

We now show that this equation is indeed satisfied by the cluster point X(·). We firstfind a sequence of X r,m(·) that converges to X(·). We then obtain

|Qi(t)− Qi(0)+∫ t

0ΛIi(Q(u))du− Di(t)|

≤ |Qi(t)−Qr,mi (t)|+ |Qi(0)−Qr,m

i (0)|

+|Di(t)−Dr,mi (t)|+ |

∫ t

0Ii(Q(u))du−

∫ t

0Ii(Qr,m(u))du|

+|Qr,mi (t)−Qr,m

i (0)+∫ t

0ΛIi(Qr,m(u))du−Dr,m

i (t)|

= |Qi(t)−Qr,mi (t)|+ |Qi(0)−Qr,m

i (0)|+ |Di(t)−Dr,mi (t)|+ |

∫ t

0Ii(Q(u))du−

∫ t

0Ii(Qr,m(u))du|.

Since Ii(·) is a continuous function, |∫ t

0 Ii(Q(u))du−∫ t

0 Ii(Qr,m(u))du| is bounded.Moreover, all other terms are bounded by construction, and therefore |Qi(t)−Qi(0)+

∫ t0 ΛIi(Q(u))du− Di(t)| → 0 as r→ ∞. Likewise, other fluid equations can

be verified as well. ut

Proof of Lemma A6

First we will show that there exists a constant r0 such that W r(s)≥ ζ − εr in proba-bility, ∀r > r0. The proof is by contradiction.We let m = argmax j∈N π j to denote the supplier with the highest static price andi /∈ argmax j∈N π j in the sequel. In the following all the inequalities refer to theinequalities “in probability,” and we omit this indication for convenience. SupposeW r(s)< ζ − εr, then

∑j 6=m

Qrj(s)

µ j< ζ − ε

r− Qrm(s)µm

≤ ζ − εr. (A48)

Now since εr is given, we can define ηr =c

2(n−2)ε

r > 0 and ηr ↓ 0 because εr ↓ 0.

If for all r we can find a pair i, j ∈N such that π j +Qr

j(s)

µ j≤ πi +

Qri (s)µi−η

r, then


letting r→ ∞ we have found a sequence that contradicts Proposition 2. Thus fromnow on we assume that

π j +Qr

j(s)

µ j> πi +

Qri (s)µi−η

r, ∀ j 6= i,m; i, j ∈N . (A49)

Combining Equations (A48) and (A49), we obtain

ζ − εr > ∑

j 6=m

Qrj(s)

µ j≥ (n−1)

Qri (s)µi

+1c ∑

j 6=i,m(πi−π j)−

n−2c

ηr

⇒ Qri (s)µi≤ 1

n−1[ζ − ε

r− 1c ∑

j 6=i,m(πi−π j)+

n−2c

ηr].

The proposed cost by supplier i is upper bounded by:

πi + cQr

i (s)µi

≤ πi +c

n−1

1

c/n

[πm−

1n ∑

k∈Nπk

]− ε

r− 1c ∑

j 6=i,m(πi−π j)+

n−2c

ηr

≤ πi +c

n−1

[nc

πm−1c ∑

k∈Nπk−

1c ∑

j 6=i,m(πi−π j)

]− c

n−1

(ε

r− n−2c

ηr)

By our choice of ηr, the last term is strictly negative. The other terms can be com-bined such that

πi + cQr

i (s)µi≤ 1

n−1[(n−1)πi +nπm− ∑

j∈Nπ j− (n−2)πi + ∑

j 6=i,mπ j]−

c2(n−1)

εr

= πm−c

2(n−1)ε

r.

Since the queue length can never be negative, we conclude that

πi + cQr

i (s)µi≤ πm + c

Qrm(s)µm

− c2(n−1)

εr, (A50)

which contradicts Proposition 2 if we let r→ ∞. since costs proposed by supplier iand m are different. Thus, W r(s)≥ ζ − εr, ∀r ≥ r0.The next thing is to show that Qr(s)> 0 in probability if W r(s)≥ ζ + εr. We againprove this by contradiction. Suppose that there exists i such that Qr

i (s) = 0. If i /∈argmax j∈N π j, then

πm + cQr

m(s)µm

≥ πm > πi = πi + cQr

i (s)µi

, (A51)

which contradicts the state space collapse while r→∞. So it suffices to consider thecase Qr

m(s) = 0.


Recall the definition of ηr and apply state space collapse, we have

π j +cQr

j(s)µ j≤ πi + c

Qri (s)µi

+ηr

c,∀ j 6= m,

⇒Qr

j(s)

µ j≤ 1

c(πi−π j)+

Qri (s)µi

+ηr

c,∀ j 6= m.

Note that Qrm(s) = 0 implies ζ + εr ≤ W r(s) = ∑ j 6=m

Qrj(s)

µ j, and therefore

ζ +εr ≤ 1

c ∑j 6=i,m

(πi−π j)+(n−1)Qr

i (s)µi

+(n−2)

cη

r,

⇒ Qri (s)µi≥ 1

n−1[ζ − 1

c ∑j 6=i,m

(πi−π j)]+ εr− (n−2)

cη

r.

The cost proposed by supplier i is lower bounded by

πi+cQr

i (s)µi≥ πi+

cn−1

[1

c/nπm−

1c/n

1n ∑

k∈Nπk−

n−2c

πi+1c ∑

j 6=i,mπ j]+c(εr− (n−2)

cη

r).

(A52)After some manipulation, the above inequality can be rewritten as

πi + cQr

i (s)µi≥ πm +

c2

εr = πm + c

Qrm(s)µm

+c2

εr, (A53)

where the last equality follows from Qrm(s) = 0. This, however, contradicts Propo-

sition 2, and henceeQr(s)> 0 if W r(s)≥ ζ + εr in probability. ut

Proof of Lemma A7

Let us define Ψ ∗(µ) = maxπ µπ−L (π) . Consider two service rates µ1 and µ2,and assume without loss of generality that µ1 > µ2. By definition the maximizer forsupplier with µ1 is π∗1 , i.e., Ψ ∗(µ1) = µ1π∗1 −L (π∗1 ). From the optimal condition,we have

Ψ∗(µ2) = max

πµ2π−L (π) ≥ µ2π

∗1 −L (π∗1 ) = µ1π

∗1 −L (π∗1 )+π

∗1 (µ2−µ1) =Ψ

∗(µ1)+π∗1 (µ2−µ1)

⇔Ψ∗(µ2)≥Ψ

∗(µ1)+π∗1 (µ2−µ1),∀µ1,µ2.

Also, from the Envelope Theorem, ∂Ψ∗(µ1)∂ µ1

= π∗1 . Note that the above inequalityholds for arbitrary pair of µ1,µ2, and hence Ψ ∗(µ) is convex in µ .


Since Ψ ∗(µ) is convex, its derivative π∗i (µ) is increasing in µ , and therefore π∗1 ≥π∗2 . The strict monotonicity follows immediately from the first-order condition ofmaxπ µπ−L (π) . Moreover, π(µ)=Ψ ∗(µ)/µ can be regarded as the average of

the derivative over [0,µ], i.e.,Ψ ∗(µ)

µ=

1µ

∫µ

0

∂Ψ(v)∂v

dv, by the strict monotonicity

of π∗(µ), it is also monotonic. Therefore, π1 := π(µ1)> π2 := π(µ2). ut

Design of an aggregated marketplace under …...a bid that comprises a ﬁxed price and a target...

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